Italian Journal of Pure and Applied Mathematicsijpam.uniud.it/online_issue/IJPAM_no-43-2020.pdf · [email protected] Giovanni Falcone Dipartimento di Metodi e Modelli Matematici

N° 43 – February 2020

Italian Journal of Pure and

Applied Mathematics

ISSN 2239-0227

EDITOR-IN-CHIEF

Piergiulio Corsini

Editorial Board

Saeid Abbasbandy

Praveen Agarwal

Bayram Ali Ersoy

Reza Ameri

Luisa Arlotti

Alireza Seyed Ashrafi

Krassimir Atanassov

Vadim Azhmyakov

Malvina Baica

Federico Bartolozzi

Rajabali Borzooei

Carlo Cecchini

Gui-Yun Chen

Domenico Nico Chillemi

Stephen Comer

Irina Cristea

Mohammad Reza Darafsheh

Bal Kishan Dass

Bijan Davvaz

Mario De Salvo

Alberto Felice De Toni

Franco Eugeni

Mostafa Eslami

Giovanni Falcone

Yuming Feng

Antonino Giambruno

Furio Honsell

Luca Iseppi

James Jantosciak

Tomas Kepka

David Kinderlehrer

Sunil Kumar

Andrzej Lasota

Violeta Leoreanu-Fotea

Maria Antonietta Lepellere

Mario Marchi

Donatella Marini

Angelo Marzollo

Antonio Maturo

Fabrizio Maturo

Sarka Hozkova-Mayerova

Vishnu Narayan Mishra

M. Reza Moghadam

Syed Tauseef Mohyud-Din

Petr Nemec

Vasile Oproiu

Livio C. Piccinini

Goffredo Pieroni

Flavio Pressacco

Sanja Jancic Rasovic

Vito Roberto

Gaetano Russo

Paolo Salmon

Maria Scafati Tallini

Kar Ping Shum

Alessandro Silva

Florentin Smarandache

Sergio Spagnolo

Stefanos Spartalis

Hari M. Srivastava

Yves Sureau

Carlo Tasso

Ioan Tofan

Aldo Ventre

Thomas Vougiouklis

Hans Weber

Shanhe Wu

Xiao-Jun Yang

Yunqiang Yin

Mohammad Mehdi Zahedi

Fabio Zanolin

Paolo Zellini

Jianming Zhan

FORUM

EDITOR-IN-CHIEF

Piergiulio Corsini

Department of Civil Engineering and Architecture

Via delle Scienze 206 - 33100 Udine, Italy [email protected]

VICE-CHIEFS

Violeta Leoreanu


MANAGING BOARD

Domenico Chillemi, CHIEF Piergiulio Corsini

Irina Cristea

Alberto Felice De Toni Furio Honsell

Violeta Leoreanu

Maria Antonietta Lepellere Elena Mocanu

Livio Piccinini

Flavio Pressacco

Luminita Teodorescu Norma Zamparo

EDITORIAL BOARD

Saeid Abbasbandy Dept. of Mathematics, Imam Khomeini International University, Ghazvin, 34149-16818, Iran [email protected] Praveen Agarwal Department of Mathematics, Anand International College of Engineering Jaipur-303012, India [email protected]

Bayram Ali Ersoy Department of Mathematics, Yildiz Technical University

34349 Beşiktaş, Istanbul, Turkey [email protected] Reza Ameri Department of Mathematics University of Tehran, Tehran, Iran [email protected]

Luisa Arlotti Department of Civil Engineering and Architecture Via delle Scienze 206 - 33100 Udine, Italy [email protected] Alireza Seyed Ashrafi Department of Pure Mathematics University of Kashan, Kāshān, Isfahan, Iran

[email protected] Krassimir Atanassov Centre of Biomedical Engineering, Bulgarian Academy of Science BL 105 Acad. G. Bontchev Str. 1113 Sofia, Bulgaria [email protected] Vadim Azhmyakov

Department of Basic Sciences, Universidad de Medellin, Medellin, Republic of Colombia [email protected] Malvina Baica University of Wisconsin-Whitewater Dept. of Mathematical and Computer Sciences Whitewater, W.I. 53190, U.S.A.

[email protected] Federico Bartolozzi Dipartimento di Matematica e Applicazioni via Archirafi 34 - 90123 Palermo, Italy [email protected] Rajabali Borzooei

Department of Mathematics Shahid Beheshti University, Tehran, Iran [email protected] Carlo Cecchini Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected] Gui-Yun Chen School of Mathematics and Statistics, Southwest University, 400715, Chongqing, China [email protected] Domenico (Nico) Chillemi Executive IT Specialist, IBM z System Software IBM Italy SpA

Via Sciangai 53 – 00144 Roma, Italy [email protected] Stephen Comer Department of Mathematics and Computer Science The Citadel, Charleston S. C. 29409, USA [email protected]

Irina Cristea CSIT, Centre for Systems and Information Technologies University of Nova Gorica Vipavska 13, Rožna Dolina, SI-5000 Nova Gorica, Slovenia [email protected]

Mohammad Reza Darafsheh School of Mathematics, College of Science

University of Tehran, Tehran, Iran [email protected] Bal Kishan Dass Department of Mathematics University of Delhi, Delhi - 110007, India [email protected]

Bijan Davvaz Department of Mathematics, Yazd University, Yazd, Iran [email protected] Mario De Salvo Dipartimento di Matematica e Informatica Viale Ferdinando Stagno d'Alcontres 31, Contrada Papardo 98166 Messina

[email protected] Alberto Felice De Toni Udine University, Rector Via Palladio 8 - 33100 Udine, Italy [email protected]

Franco Eugeni Dipartimento di Metodi Quantitativi per l'Economia del Territorio Università di Teramo, Italy [email protected]

Mostafa Eslami Department of Mathematics Faculty of Mathematical Sciences University of Mazandaran, Babolsar, Iran [email protected] Giovanni Falcone Dipartimento di Metodi e Modelli Matematici

viale delle Scienze Ed. 8 90128 Palermo, Italy [email protected] Yuming Feng College of Math. and Comp. Science, Chongqing Three-Gorges University, Wanzhou, Chongqing, 404000, P.R.China [email protected]

Antonino Giambruno Dipartimento di Matematica e Applicazioni via Archirafi 34 - 90123 Palermo, Italy [email protected] Furio Honsell Dipartimento di Matematica e Informatica

Via delle Scienze 206 - 33100 Udine, Italy [email protected] Luca Iseppi Department of Civil Engineering and Architecture, section of Economics and Landscape Via delle Scienze 206 - 33100 Udine, Italy [email protected]

James Jantosciak Department of Mathematics, Brooklyn College (CUNY) Brooklyn, New York 11210, USA [email protected] Tomas Kepka MFF-UK Sokolovská 83

18600 Praha 8,Czech Republic [email protected] David Kinderlehrer Department of Mathematical Sciences, Carnegie Mellon University Pittsburgh, PA15213-3890, USA [email protected]

Sunil Kumar Department of Mathematics, National Institute of Technology Jamshedpur, 831014, Jharkhand, India [email protected] Andrzej Lasota Silesian University, Institute of Mathematics Bankova 14

40-007 Katowice, Poland [email protected] Violeta Leoreanu-Fotea Faculty of Mathematics Al. I. Cuza University 6600 Iasi, Romania [email protected]

Maria Antonietta Lepellere Department of Civil Engineering and Architecture Via delle Scienze 206 - 33100 Udine, Italy [email protected] Mario Marchi Università Cattolica del Sacro Cuore via Trieste 17, 25121 Brescia, Italy

[email protected] Donatella Marini Dipartimento di Matematica Via Ferrata 1- 27100 Pavia, Italy [email protected] Angelo Marzollo

Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected] Antonio Maturo University of Chieti-Pescara, Department of Social Sciences, Via dei Vestini, 31 66013 Chieti, Italy

[email protected] Fabrizio Maturo University of Chieti-Pescara, Department of Management and Business Administration, Viale Pindaro, 44 65127 Pescara, Italy [email protected]

Sarka Hoskova-Mayerova Department of Mathematics and Physics University of Defence Kounicova 65, 662 10 Brno, Czech Republic [email protected] Vishnu Narayan Mishra Applied Mathematics and Humanities Department

Sardar Vallabhbhai National Institute of Technology 395 007, Surat, Gujarat, India [email protected] M. Reza Moghadam Faculty of Mathematical Science, Ferdowsi University of Mashhadh P.O.Box 1159 - 91775 Mashhad, Iran

[email protected] Syed Tauseef Mohyud-Din Faculty of Sciences, HITEC University Taxila Cantt Pakistan [email protected] Petr Nemec Czech University of Life Sciences, Kamycka’ 129 16521 Praha 6, Czech Republic [email protected] Vasile Oproiu Faculty of Mathematics Al. I. Cuza University 6600 Iasi, Romania [email protected] Livio C. Piccinini Department of Civil Engineering and Architecture Via delle Scienze 206 - 33100 Udine, Italy [email protected] Goffredo Pieroni Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy

[email protected]

Flavio Pressacco Dept. of Economy and Statistics Via Tomadini 30 33100, Udine, Italy [email protected] Sanja Jancic Rasovic Department of Mathematics Faculty of Natural Sciences and Mathematics, University of Montenegro Cetinjska 2 – 81000 Podgorica, Montenegro [email protected] Vito Roberto

Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected] Gaetano Russo Department of Civil Engineering and Architecture Via delle Scienze 206 33100 Udine, Italy

[email protected] Paolo Salmon Dipartimento di Matematica, Università di Bologna Piazza di Porta S. Donato 5 40126 Bologna, Italy [email protected]

Maria Scafati Tallini Dipartimento di Matematica "Guido Castelnuovo" Università La Sapienza Piazzale Aldo Moro 2 - 00185 Roma, Italy [email protected] Kar Ping Shum Faculty of Science The Chinese University of Hong Kong

Hong Kong, China (SAR) [email protected] Alessandro Silva Dipartimento di Matematica "Guido Castelnuovo", Università La Sapienza Piazzale Aldo Moro 2 - 00185 Roma, Italy [email protected]

Florentin Smarandache Department of Mathematics, University of New Mexico Gallup, NM 87301, USA [email protected] Sergio Spagnolo Scuola Normale Superiore Piazza dei Cavalieri 7 - 56100 Pisa, Italy

[email protected] Stefanos Spartalis Department of Production Engineering and Management, School of Engineering, Democritus University of Thrace V.Sofias 12, Prokat, Bdg A1, Office 308 67100 Xanthi, Greece [email protected] Hari M. Srivastava Department of Mathematics and Statistics University of Victoria, Victoria, British Columbia V8W3P4, Canada [email protected] Yves Sureau 27, rue d'Aubiere

63170 Perignat, Les Sarlieve - France [email protected] Carlo Tasso Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Ioan Tofan Faculty of Mathematics Al. I. Cuza University 6600 Iasi, Romania [email protected] Aldo Ventre Seconda Università di Napoli, Fac. Architettura, Dip. Cultura del Progetto

Via San Lorenzo s/n 81031 Aversa (NA), Italy [email protected] Thomas Vougiouklis Democritus University of Thrace, School of Education, 681 00 Alexandroupolis. Greece

[email protected] Hans Weber Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected] Shanhe Wu Department of Mathematics, Longyan University,

Longyan, Fujian, 364012, China [email protected] Xiao-Jun Yang Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu, 221008, China [email protected]

Yunqiang Yin School of Mathematics and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, P.R. China [email protected]

Mohammad Mehdi Zahedi

Department of Mathematics, Faculty of Science Shahid Bahonar, University of Kerman Kerman, Iran [email protected] Fabio Zanolin Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy

[email protected] Paolo Zellini Dipartimento di Matematica, Università degli Studi Tor Vergata via Orazio Raimondo (loc. La Romanina) 00173 Roma, Italy [email protected]

Jianming Zhan Department of Mathematics, Hubei Institute for Nationalities Enshi, Hubei Province,445000, China [email protected]

mailto:[email protected]:[email protected]:[email protected]:[email protected]

i ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 43-2020

In memoriam of Professor Ivo Rosenberg

The “Italian Journal of Pure and Applied Mathematics“ cannot more take advantage of the

precious collaboration of prof. Ivo Rosenberg, who has passed away.

The members of Editorial Board express their deep sorrow for this loss.

The Chief Editor regrets the loss of Prof. Ivo Rosenberg. He has been a great man of

science and a very dear friend.

All they who knew him will remember always his scientific value and his exquisite human

qualities.

Piergiulio Corsini

ii ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 43-2020

iii ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 43-2020

Italian Journal of Pure and Applied Mathematics ISSN 2239-0227

Web Site

http://ijpam.uniud.it/journal/home.html

Twitter @ijpamitaly

https://twitter.com/ijpamitaly

EDITOR-IN-CHIEF

Piergiulio Corsini

Department of Civil Engineering and Architecture

Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Vice-CHIEFS

Violeta Leoreanu-Fotea Maria Antonietta Lepellere

Managing Board

Domenico Chillemi, CHIEF Piergiulio Corsini

Irina Cristea

Alberto Felice De Toni

Furio Honsell

Violeta Leoreanu-Fotea


Elena Mocanu

Livio Piccinini Flavio Pressacco

Luminita Teodorescu

Norma Zamparo

Editorial Board

Saeid Abbasbandy

Praveen Agarwal Bayram Ali Ersoy

Reza Ameri

Luisa Arlotti Alireza Seyed Ashrafi

Krassimir Atanassov

Vadim Azhmyakov Malvina Baica

Federico Bartolozzi

Rajabali Borzooei Carlo Cecchini

Gui-Yun Chen

Domenico Nico Chillemi Stephen Comer

Irina Cristea

Mohammad Reza Darafsheh Bal Kishan Dass

Bijan Davvaz

Mario De Salvo Alberto Felice De Toni

Franco Eugeni

Mostafa Eslami Giovanni Falcone

Yuming Feng

Antonino Giambruno

Furio Honsell Luca Iseppi

James Jantosciak

Tomas Kepka David Kinderlehrer

Sunil Kumar

Andrzej Lasota Violeta Leoreanu-Fotea


Mario Marchi Donatella Marini

Angelo Marzollo

Antonio Maturo Fabrizio Maturo

Sarka Hozkova-Mayerova

Vishnu Narayan Mishra M. Reza Moghadam

Syed Tauseef Mohyud-Din

Petr Nemec Vasile Oproiu

Livio C. Piccinini

Goffredo Pieroni Flavio Pressacco

Sanja Jancic Rasovic

Vito Roberto

Gaetano Russo Paolo Salmon

Maria Scafati Tallini

Kar Ping Shum Alessandro Silva

Florentin Smarandache

Sergio Spagnolo Stefanos Spartalis

Hari M. Srivastava

Yves Sureau Carlo Tasso

Ioan Tofan

Aldo Ventre Thomas Vougiouklis

Hans Weber

Shanhe Wu Xiao-Jun Yang

Yunqiang Yin

Mohammad Mehdi Zahedi Fabio Zanolin

Paolo Zellini

Jianming Zhan

Forum Editrice Universitaria Udinese Srl

Via Larga 38 - 33100 Udine

Tel: +39-0432-26001, Fax: +39-0432-296756 [email protected]

http://ijpam.uniud.it/journal/home.htmlhttp://ijpam.uniud.it/journal/home.htmlmailto:[email protected]:[email protected]

ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 43–2020 iv

Table of contents

Arslan Hojat Ansari, Kamaleldin AbodayehUpper class functions on a controlled contraction principle in

partial S-metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–13

Yanheng Chen, Guiyun Chen, Yuming Feng, B.O. OnasanyaCharacterization of some linear groups by their conjugacy class sizes . . . . . . . . . . . . . . . . . . . 14–24

M. Lellis Thivagar, Saeid Jafari, V. Sutha DeviOn new class of contra continuity in nano topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–36

Song-Tao Guo, Yong Xu, Guiyun ChenHeptavalent symmetric graphs of order 8p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37–46

Kanak Kanti BaishyaOn almost generalized pseudo-Ricci symmetric spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47–54

Masoud Zolfaghari, Mohammad Hosein Moslemi KoopaeiSome remarks on generalizations of prime submodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55–64

Majid Mohammed AbedA new view of closed-CS-module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65–72

A.F. Talee, M.Y. Abbasi, G. Muhiuddin, Sabahat Ali KhanHesitant fuzzy sets approach to ideal theory in ordered Γ-semigroups . . . . . . . . . . . . . . . . . . . 73–85

R. Dhavaseelan, S. JafariIntuitionistic fuzzy rare α-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86–94

Shireen O. DakheelEn-semi prime subacts over monoids with zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95–99

G. Muhiuddin, Shuaa AldhafeeriJoin hesitant fuzzy filters of residuated lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100–114

J. Alebraheem, S. Hussain, F. Ahmad, K. NimerApplication of differential transformation method for solving

prey predator model with holling type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115–127

Aydah Mohammed Ayed Al-AhmadiSome classes of mappings on generalized quaternion metric spaces II . . . . . . . . . . . . . . . . 128–140

Farzaneh Nowroozi Larki, Shahram Rayat PishehOn two-sided group digraphs and graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141–152

Shao Changguo, Jiang Qinhui, Li KefengA characterization of simple group S4(7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153–157

Xiaohong Zhang, Peng Yu, F. Smarandache, Choonkil ParkRedefined neutrosophic filters in BE-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158–176

Zuhier AltawallbehMore on almost countably compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177–184

Anwar Bataihah, Abdalla Tallafha, Wasfi ShatanawiFixed point results with Ω-distance by utilizing simulation functions . . . . . . . . . . . . . . . . . .185–196

Amol B. Patil, Uday H. NaikOn coefficient inequalities for certain subclasses of meromorphic

bi-univalent functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197–205

ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 43-2020 v

Junjian YangA note on unitarily invariant norm inequalities for

accretive-dissipative operator matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206–212

Guowei ZhangLimiting direction of Julia sets and infinite radial order of

solutions to complex linear differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213–223

H. Jude Immaculate, Saeid Jafari, I. ArockiaraniIntuitionistic fuzzy ideals on approximation systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224–236

Talal Al-HawaryOn modular flats and pushouts of matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237–241

S.T. Mohyud-Din, H. Khan, M. Arif, S. BushnaqChebyshev wavelet method (CWM) for the numerical solutions of

fractional boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242–255

S. Dhanasekar, S. Hariharan, David Maxim GururajFuzzy zero suffix algorithm to solve fully fuzzy transportation

problems by using element-wise operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .256–267

Sang Wu, Chao Xu, Yinghui DongPricing European call options with default risk under a

jump-diffusion model via FFT transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268–278

Chao WeiEstimation for the parameter of a class of diffusion processes . . . . . . . . . . . . . . . . . . . . . . . . 279–290

Umar Hayat, Ghulam Farid, Erdal KarapinarRelationships between tropical eigenvectors and tropical

fixed points of the group GL(2,R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291–300S. Boughaba, M. El Amine Bencheikh Le Hocine, M. HaiourL∞-asymptotic behavior for a finite element approximation

to optimal control problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301–312

A. MolkhasiRings with strongly algebraically closed lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313–319

N. Fahimi, T. Nozari, R. MahjoobOn the n∗− and γ∗n− complete fuzzy hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320–330F.M. El-Sabaa, M. Hosny, S.K. ZakriaBifurcations of Liouville tori of generalized two-fixed

center problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331–352

Ghaleb GumahOptimization technique for solving fuzzy partial differential

equations under strongly generalized differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353–379

A.H. El-Qadeem, D.A. MohanOn some properties of certain subclasses of univalent functions . . . . . . . . . . . . . . . . . . . . . . 380–390

M. AnbarloeiOn uniformly primary hyperideals and uniformly 2-absorbing

primary hyperideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391–401

A. El-Sayed Ahmed, M.Y. YoussifClasses of weighted tent function spaces and mixed norms

with some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402–415

Shatha Hasan, Mona SakkijhaToward strictly singular fractional operator restricted by

Fredholm-Volterra in Sobolev space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416–427

vi ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 43-2020

Lin Chen, Jun Li, Jianhua ZhangNonlinear left ∗-Lie triple mappings of standard

operator algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .428–438

Deepti ThakurA common fixed point theorem without continuity under weak compatible

mappings in uniform convex Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439–446

Jian Tang, Ze Gu, Xiangyun XieA further study on the hyperideals of ordered semihypergroups . . . . . . . . . . . . . . . . . . . . . . . 447–470

Lü Gong, Libo ZhaoFinite group with coincide automizer and central

automorphism of subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471–476

Aysel Turgut Vanli, Inan UnalRicci semi-symmetric normal complex contact metric manifolds . . . . . . . . . . . . . . . . . . . . . 477–491

Eman Hussein, Hasan Al-Ezeh, Omar Abu GhneimThe path graph of the amalgamated graph of C3 and Cn

at an edge or at a vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492–502

Shitian Liu, Xianhua LiA characterization of some alternating group by its order and

special conjugacy class sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503–516

N.R. Abed Alhaleem, A.H. HandamInvo-clean rings associated with central polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517–522

Maryam Attary, Praveen AgarwalOn developing an optimal Jarratt-like class for

solving nonlinear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .523–530

A. Tajmouati, M. Amouch, M.R.F. Alhomidi ZakariyaSome spectral inclusion for strongly continuous

semigroups operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .531–543

Khaled Moaddy, A. AlhashashA reliable non-standard finite difference scheme for solving nonlinear

biochemical reaction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544–551

Areerat Vongkok, Nopparat PochaiNumerical simulations for reactive nitrogen compounds pollution

measurements in a stream using Saulyev method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552–582

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contraction of Ciric-Berinde type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653–670

ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 43-2020 vii

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viii ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 43-2020

Germina K. Augusthy, P. SooryaVertex (n, k)-choosability of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905–911

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ixITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 43-2020

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ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 43–2020 (1–13) 1

Upper class functions on a controlled contraction principle inpartial S-metric spaces

Arslan Hojat Ansari∗

Department of MathematicsKaraj BranchIslamic Azad [email protected]

Kamaleldin AbodayehDepartment of Mathematical Sciences

Prince Sultan University

Riyadh, 11586

Saudi Arabia

[email protected]

Abstract. In this paper, we prove the existence and uniqueness of a fixed point of aself mapping on partial S-metric spaces under the partially α-contractive condition.

Keywords: common fixed point.

1. Introduction and mathematical preliminaries

The existence and uniqueness of a fixed point for a self mapping on differenttypes of metric spaces were the main topic for many research papers [4-18].The notion of S-metric space was introduced by Sedghi [3]. A generalization ofS-metric space was given by Nabil in [1], where he introduced partial S-metricspaces. Moreover, he proved the existence of a fixed point for a self mapping inpartial S-metric space. In this paper, we generalize the results in [1] by addinga control function to the contraction principle, which makes the results in [1] adirect consequences of our theorems.

Before proceeding to the main results, we set forth some definitions that willbe used in the sequel.

Definition 1.1 ([4]). Let X be a nonempty set and p : X×X −→ [0,+∞). Wesay that (X, p) is a partial metric space if for all x, y, z ∈ X we have:

1. x = y if and only if p(x, y) = p(x, x) = p(y, y);

2. p(x, x) ≤ p(x, y);

∗. Corresponding author

2 ARSLAN HOJAT ANSARI and KAMALELDIN ABODAYEH

3. p(x, y) = p(y, x);

4. p(x, z) ≤ p(x, y) + p(y, z)− p(y, y).

Definition 1.2 ([3]). Let X be a nonempty set. An S-metric space on Xis a function S : X3 → [0,∞) that satisfies the following conditions, for allx, y, z, a ∈ X :

1. S(x; y; z) ≥ 0,

2. S(x; y; z) = 0 if and only if x = y = z,

3. S(x; y; z) ≤ S(x;x; a) + S(y; y; a) + S(z; z; a).

The pair (X;S) is called an S-metric space.

Next, we give the definition of partial S-metric space.

Definition 1.3 ([1]). Let X be a nonempty set. A partial S-metric space onX is a function Sp : X

3 → [0,∞) that satisfies the following conditions, for allx, y, z, t ∈ X :

(i) x = y if and only if Sp(x, x, x) = Sp(y, y, y) = Sp(x, x, y);

(ii) Sp(x, y, z) ≤ Sp(x, x, t) + Sp(y, y, t) + Sp(z, z, t)− Sp(t, t, t);(iii) Sp(x, x, x) ≤ Sp(x, y, z);(iv) Sp(x, x, y) = Sp(y, y, x).

The pair (X,Sp) is called a partial S-metric space.

Definition 1.4. A sequence {xn}∞n=0 of elements in (X,Sp) is called p-Cauchy ifthe limit limn,m→∞ Sp(xn, xn, xm) exists and finite. The partial S-metric space(X,Sp) is called complete if for each p-Cauchy sequence {xn}∞n=0 there existsz ∈ X such that Sp(z, z, z) = limn Sp(z, z, xn) = limn,m Sp(xn, xn, xm).

Moreover, (X,Sp) is a complete partial S-metric space if and only if (X,Sp)is a complete S-metric space. A sequence {xn}n in a partial S-metric space(X,Sp) is called 0-Cauchy if limn,m→∞ Sp(xn, xn, xm) = 0. We say that (X,Sp)is 0-complete if every 0-Cauchy in X converges to a point x ∈ X such thatSp(x, x, x) = 0.

One can easily construct an example of a partial S-metric space by using theordinary partial metric space.

Example 1.5 ([1]). Let X = [0,∞) and p be the ordinary partial metric spaceon X. Define the mapping on X3 to be Sp(x, y, z) = p(x, z) + p(y, z). Then Spdefines a partial S-metric space.

Now we introduce the notion of partially α−contractive.

UPPER CLASS FUNCTIONS ON A CONTROLLED CONTRACTION PRINCIPLE ... 3

Definition 1.6. Let (X,Sp) be a partial S-metric space and T : X −→ X bea given mapping. We say that T is partially α−contractive if there exists aconstant k ∈ [0, 1) and a function α : X × X −→ [0,+∞) such that for allx, y ∈ X we have

(1.1) α(x, y)Sp(Tx, Tx, Ty) ≤ max{kSp(x, x, y), Sp(x, x, x), Sp(y, y, y)}.

Definition 1.7. Let (X,Sp) be a partial S-metric space and T : X −→ X be agiven mapping. We say that T is Rα−admissible if x, y ∈ X, α(x, y) ≥ 1 impliesthat α(x, Ty) ≥ 1. Also, we say that T is α−admissible if x, y ∈ X, α(x, y) ≥ 1implies that α(Tx, Ty) ≥ 1.

Example 1.8. Let X = [0,+∞). Define T : X −→ X by Tx =√x and

α : X ×X −→ X by

α(x, y) =

{ex−y, if x ≥ y0, if x < y.

It is a straightforward to verify that T is α−admissible and Rα−admissible.

Now, we set

ρSp(α) : = inf{Sp(x, x, y) | x, y ∈ X : α(x, y) ≥ 1}= inf{Sp(x, x, x) | x ∈ X : α(x, x) ≥ 1},XSp(α) = {x ∈ X | Sp(x, x, x) = ρSp(α)},ZSp(α) = {x ∈ XSp | α(x, x) ≥ 1}.

Definition 1.9. Let (X,Sp) be a partial S-metric space and T : X −→ X bea given mapping. We say that T is Rµ−subadmissible if x, y ∈ X, µ(x, y) ≤ 1implies that µ(x, Ty) ≤ 1.

2. Main result

In this section, we prove the existence of a fixed point in a partial S-metricspace. We prove relevant corollary. This next theorem is considered to be ourmain result.

Definition 2.1 ([?]). Let T : X → X be a map and µ : X ×X → [0,+∞) be afunction. We say that T is µ-subadmissible if x, y ∈ X, µ(x, y) ≤ 1 implies thatµ(Tx, Ty) ≤ 1.

Definition 2.2. A map T : X → X is said to be triangular µ-subadmissible ifthe following holds:

(T1) T is µ-subadmissible,

(T2) µ(x, u) ≤ 1 and µ(u, y) ≤ 1 implies that µ(x, y) ≤ 1, x, u, y ∈ X.


Lemma 2.3. Let T : X → X be a triangular µ− suborbital admissible map-ping. Assume that there exists x1 ∈ X such that µ (x1, Tx1) ≤ 1. Then thereexists a sequence {xn} such that µ (xn, xm) ≤ 1 for all m, n ∈ N with n < m.

The letter N represent the set of positive integers and N0 = N∪{0}. Further,the nonnegative real numbers will be denoted by R+0 = [0,∞).

In 2014 the concept of pair (F , h) is an upper class (see Definition 2.4 until2.10 ) was introduced by A.H. Ansari in [19]

Definition 2.4 ([19, 20]). A function h : R+×R+ → R is said to be a functionof subclass of type I, if x ≥ 1 =⇒ h(1, y) ≤ h(x, y) for all y ∈ R+.

Example 2.5 ([19, 20]). Define h : R+ × R+ → R by:

(a) h(x, y) = (y + l)x, l > 1;

(b) h(x, y) = (x+ l)y, l > 1;

(c) h(x, y) = xny, n ∈ N;

(d) h(x, y) = y;

(e) h(x, y) = 1n+1(∑n

i=0 xi)y, n ∈ N;

(f) h(x, y) =[

1n+1

(∑ni=0 x

i)+ l]y, l > 1, n ∈ N

for all x, y ∈ R+. Then h is a function of subclass of type I.

Definition 2.6 ([19, 20]). Let h,F : R+ × R+ → R, then we say that the pair(F , h) is an upper class of type I, if h is a function of subclass of type I and: (i)0 ≤ s ≤ 1 =⇒ F(s, t) ≤ F(1, t), (ii) h(1, y) ≤ F(1, t) =⇒ y ≤ t for all t, y ∈ R+.

Example 2.7 ([19, 20]). Define h,F : R+ × R+ → R by:

(a) h(x, y) = (y + l)x, l > 1 and F(s, t) = st+ l;

(b) h(x, y) = (x+ l)y, l > 1 and F(s, t) = (1 + l)st;

(c) h(x, y) = xmy, m ∈ N and F(s, t) = st;

(d) h(x, y) = y and F(s, t) = t;

(d) h(x, y) = 1n+1(∑n

i=0 xi)y, n ∈ N and F(s, t) = st;

(e) h(x, y) =[

1n+1

(∑ni=0 x

i)+ l]y, l > 1, n ∈ N and F(s, t) = (1 + l)st

for all x, y, s, t ∈ R+. Then the pair (F , h) is an upper class of type I.


Definition 2.8 ([19, 20]). A function h : R+ × R+ × R+ → R is said to be afunction of subclass of type II, if for all x, y ≥ 1, we have h(1, 1, z) ≤ h(x, y, z),for all z ∈ R+.

Example 2.9 ([19, 20]). Define h : R+ × R+ × R+ → R by:

(a) h(x, y, z) = (z + l)xy, l > 1;

(b) h(x, y, z) = (xy + l)z, l > 1;

(c) h(x, y, z) = z;

(d) h(x, y, z) = xmynzp,m, n, p ∈ N;

(e) h(x, y, z) = xm+xnyp+yq

3 zk,m, n, p, q, k ∈ N

for all x, y, z ∈ R+. Then h is a function of subclass of type II.

Definition 2.10 ([19, 20]). Let h : R+ ×R+ ×R+ → R and F : R+ ×R+ → R,then we say that the pair (F , h) is an upper class of type II, if h is a subclass oftype II and the following holds:

(i) if 0 ≤ s ≤ 1 then we have F(s, t) ≤ F(1, t),

(ii) if h(1, 1, z) ≤ F(s, t) then we have z ≤ st for all s, t, z ∈ R+.

Example 2.11 ([19, 20]). Define h : R+×R+×R+ → R and F : R+×R+ → Rby:

(a) h(x, y, z) = (z + l)xy, l > 1,F(s, t) = st+ l;

(b) h(x, y, z) = (xy + l)z, l > 1,F(s, t) = (1 + l)st;

(c) h(x, y, z) = z, F (s, t) = st;

(d) h(x, y, z) = xmynzp,m, n, p ∈ N,F(s, t) = sptp

(e) h(x, y, z) = xm+xnyp+yq

3 zk,m, n, p, q, k ∈ N,F(s, t) = sktk

for all x, y, z, s, t ∈ R+. Then the pair (F , h) is an upper class of type II.

Notation.

ρSp(α, µ) := inf{Sp(x, x, y) | x, y ∈ X : α(x, y) ≥ 1, µ(x, y) ≤ 1}= inf{Sp(x, x, x) | x ∈ X : α(x, x) ≥ 1, α(x, x) ≤ 1},

XSp(α, µ) = {x ∈ X | Sp(x, x, x) = ρSp(α, µ)},ZSp(α, µ) = {x ∈ XSp | α(x, x) ≥ 1, µ(x, x) ≤ 1}.


Definition 2.12. Let (X,Sp) be a partial S-metric space and T : X −→ X bea given mapping. We say that T is partially (F , h, α, µ)-contractive if thereexists a constant k ∈ [0, 1) and a function α, µ : X ×X −→ [0,+∞) such thatfor all x, y ∈ X we have

h(α(x, y), Sp(Tx, Tx, Ty)) ≤ F(µ(x, y),max{kSp(x, x, y),Sp(x, x, x), Sp(y, y, y)}).(2.1)

where the pair (F , h) is an upper class of type I.

Theorem 2.13. Let (X,Sp) be a complete partial S-metric space, T be a selfmapping on X and assume that T is partially (F , h, α, µ)-contractive. If Tis α−admissible ,µ−subadmissible and Rα−admissible, Rµ−subadmissible andif XSp(α, µ) is nonempty, then ZSp(α, µ) is nonempty. Also, assume that thereexists x0 ∈ X such that α(x0, x0) ≥ 1, µ(x0, x0) ≤ 1, then there exists a ∈ ZSp(α)such that Ta = a.

Moreover, if for all u, v in ZSp(α, µ) with the property Tu = u and Tv = vwe have α(u, v) ≥ 1, µ(u, v) ≤ 1, then T has a unique fixed point in ZSp(α, µ).

Proof. Let x0 ∈ X such that α(x0, x0) ≥ 1. Define a sequence {xn} for alln ≥ 0 in X such that x1 = Tx0, x2 = Tx1, · · · , xn+1 = Txn, · · · . Since Tis α−admissible ,µ−subadmissible and Rα−admissible, Rµ−subadmissible, wehave α(x0, x1) = α(x0, Tx0) ≥ 1 ,µ(x0, x1) = µ(x0, Tx0) ≤ 1 , and henceα(x1, x2) = α(Tx0, Tx1) ≥ 1, µ(x1, x2) = µ(Tx0, Tx1) ≤ 1 . So, by inductionon n we get

α(xn, xn+1) ≥ 1, µ(xn, xn+1)) ≤ 1,

for all n ≥ 0. Also, since T is Rα−admissible and Rµ−subadmissible; α(x0, x0) ≥1, µ(x0, x0) ≤ 1 implies α(x0, x1) = α(x0, Tx0) ≥ 1, µ(x0, x1) = µ(x0, Tx0) ≤ 1.By induction on n, we also conclude that

α(x0, xn) ≥ 1, µ(x0, xn) ≤ 1

for all n ≥ 0. Also, given the fact that T is α-admissible and α(x0, x0) ≥ 1, itnot difficult to prove that α(xn, xn) ≥ 1 for all n ≥ 0. Hence,

h(1, Sp(x1, x1, x1) = h(1, Sp(Tx0, Tx0, Tx0))

≤ h(α(x0, x0), Sp(Tx0, Tx0, Tx0))≤ F(µ(x0, x0),max{kSp(x0, x0, x0), Sp(x0, x0, x0), Sp(x0, x0, x0)})≤ F(1,max{kSp(x0, x0, x0), Sp(x0, x0, x0), Sp(x0, x0, x0)}).

This implies that

Sp(x1, x1, x1) ≤ max{kSp(x0, x0, x0), Sp(x0, x0, x0), Sp(x0, x0, x0)}= Sp(x0, x0, x0).


By induction on n, we obtain:

Sp(xn+1, xn+1, xn+1) ≤ Sp(xn, xn, xn).

Therefore, {Sp(xn, xn, xn)}{n≥0} is a nonincreasing sequence. Define

r0 := limnSp(xn, xn, xn) = inf

nSp(xn, xn, xn) ≥ 0

and

M0 :=2

1− kSp(x0, x0, x1) + Sp(x0, x0, x0).

Next, we show that Sp(x0, x0, xn) ≤ M0, for any n ≥ 0. If n = 0; thecase is trivial. For n = 1 and using the fact that k ∈ [0, 1) we deduce thatSp(x0, x0, x1) ≤

2

1− kSp(x0, x0, x1) ≤

2

1− kSp(x0, x0, x1)+Sp(x0, x0, x0) =M0.

So, we may assume that it is true for all n ≤ n0−1 and prove it for n = n0 ≥ 2.

Sp(x0, x0, xn0) ≤ Sp(x0, x0, x1) + Sp(x0, x0, x1) + Sp(xn0 , xn0 , x1)− Sp(x1, x1, x1)≤ 2Sp(x0, x0, x1) + Sp(x1, x1, xn0)≤ 2Sp(x0, x0, x1) + α(x0, xn0−1)Sp(Tx0, Tx0, Txn0−1)≤ 2Sp(x0, x0, x1)+max{kSp(x0, x0, xn0−1), Sp(x0, x0, x0), Sp(xn0−1, xn0−1, xn0−1)}≤ 2Sp(x0, x0, x1) + max{kSp(x0, x0, xn0−1), Sp(x0, x0, x0)}.

Also, by induction assumption, we have Sp(x0, x0, xn0−1) ≤ 21−kSp(x0, x0, x1) +Sp(x0, x0, x0). So, we have

Sp(x0, x0, xn0) ≤ 2Sp(x0, x0, x1)

+ max{ 2k1− k

Sp(x0, x0, x1) + kSp(x0, x0, x0), Sp(x0, x0, x0)}

≤ 2Sp(x0, x0, x1) +2k

1− kSp(x0, x0, x1) + Sp(x0, x0, x0)

=2

1− kSp(x0, x0, x1) + Sp(x0, x0, x0) =M0.

Hence, we conclude that Sp(x0, x0, xn) ≤M0. Next, we need to show that

limn,m

Sp(xn, xn, xm) = r0.

For all n,m we have Sp(xn, xn, xm) ≥ Sp(xn, xn, xn) ≥ r0. Let ϵ > 0 find anatural number n0 such that Sp(xn0 , xn0 , xn0) < r0 + ϵ and 2M0k

n0 < r0 + ϵ.Now for any n,m ≥ 2n0, since T is Rα−admissible and using the fact thatα(xn, xn+1) ≥ 1, µ(xn, xn+1) ≤ 1 we deduce that α(xn, xm) ≥ 1, µ(xn, xm) ≤ 1.Hence,

h(1, Sp(xn, xn, xm) ≤ h(α(xn, xm), Sp(xn, xn, xm)) ≤ F(µ(xn, xm), θ) ≤ F(1, θ),


where

θ = max{kSp(xn−1, xn−1, xm−1), Sp(xn−1, xn−1, xn−1), Sp(xm−1, xm−1, xm−1)}.

This implies that

Sp(xn, xn, xm)

≤ max{kSp(xn−1, xn−1, xm−1), Sp(xn−1, xn−1, xn−1), Sp(xm−1, xm−1, xm−1)}≤ max{k2Sp(xn−2, xn−2, xm−2), Sp(xn−2, xn−2, xn−2), Sp(xm−2, xm−2, xm−2)}≤ · · · ≤ max{kn0Sp(xn−n0 , xn−n0 , xm−n0), Sp(xn−n0 , xn−n0 , xn−n0),Sp(xm−n0 , xm−n0 , xm−n0)}≤ r0 + ϵ.

Hence,limn,m

Sp(xn, xn, xm) = r0.

Since (X, p) is a complete partial S-metric space; there exists x̃ ∈ X such that

r0 = Sp(x̃, x̃, x̃) = limnSp(x̃, x̃, xn) = lim

n,mSp(xn, xn, xm).

Now, we show that Sp(x̃, x̃, x̃) = Sp(x̃, x̃, T x̃). For each natural number n wehave

Sp(x̃, x̃, T x̃) ≤ 2Sp(x̃, x̃, xn)− Sp(xn, xn, xn) + Sp(T x̃, T x̃, xn).

Using the property that T is α−contractive, we deduce that there exists a sub-sequence of natural numbers {nl} such that

h(1, Sp(T x̃, T x̃, xnl) ≤ h(α(x̃, xnl−1), Sp(T x̃, T x̃, xnl))≤ F(µ(x̃, xnl−1),max{kSp(x̃, x̃, xnl−1), Sp(x̃, x̃, x̃), Sp(xnl−1, xnl−1, xnl−1)})≤ F(1,max{kSp(x̃, x̃, xnl−1), Sp(x̃, x̃, x̃), Sp(xnl−1, xnl−1, xnl−1)}),

and thus

Sp(T x̃, T x̃, xnl) ≤ max{kSp(x̃, x̃, xnl−1), Sp(x̃, x̃, x̃), Sp(xnl−1, xnl−1, xnl−1)}.

So, for l ≥ 1, we have either Sp(T x̃, T x̃, xnl) ≤ kSp(x̃, x̃, xnl−1) or less than orequal Sp(x̃, x̃, x̃) or less than or equal Sp(xnl−1, xnl−1, xnl−1). In all of these threecases, if we take the limit as l goes toward ∞ we get Sp(x̃, x̃, T x̃) ≤ Sp(x̃, x̃, x̃).But, we know by the property (ii) of the partial S-metric space definition thatSp(x̃, x̃, x̃) ≤ Sp(x̃, x̃, T x̃). Therefore, Sp(x̃, x̃, x̃) = Sp(x̃, x̃, T x̃).

Now, we show that XSp(α, µ) is nonempty. For each natural number l pickxl ∈ X with α(xl, xl) ≥ 1 and Sp(xl, xl, xl) < ρSp(α, µ) + 1l and show that

limn,m

Sp(x̃n, x̃n, x̃m) = ρSp(α, µ).


Let ϵ > 0 put n0 := (3

ϵ(1−k))+1 if l ≥ n0 then we have: ρSp(α, µ) ≤ Sp(x̃l, x̃l, T x̃l)≤ Sp(x̃l, x̃l, T x̃l) ≤ rxl ≤ Sp(x̃l, x̃l, T x̃l) < ρSp(α, µ) + 1l ≤ ρSp(α, µ) +

1n0

<

ρSp(α, µ) +ϵ(1−k)

3 . Hence, we deduce that:

Ul := Sp(x̃l, x̃l, x̃l)− Sp(T x̃l, T x̃l, T x̃l) <ϵ(1− k)

3,

for i ≥ n0.Also, if l ≥ n0, then Sp(x̃l, x̃l, x̃l) = rxl ≤ Sp(xl, xl, xl) < ρSp(α) + 1n0 .

Which implies that Sp(x̃l, x̃l, x̃l) ≤ ρSp(α, µ) +ϵ(1−k)

3 for all l ≥ n0. Now,if n,m ≥ n0, then Sp(x̃n, x̃n, x̃m) ≤ 2Sp(x̃n, x̃n, T x̃n) + Sp(T x̃n, T x̃n, T x̃m) +2Sp(T x̃m, T x̃m, x̃m)− Sp(T x̃n, T x̃n, T x̃n)− Sp(T x̃m, T x̃m, T x̃m).

We know that Sp(x̃, x̃, x̃) = Sp(x̃, x̃, T x̃) which implies that

h(1, Sp(T x̃n, T x̃n, T x̃m) ≤ h(α(x̃n, x̃m), Sp(T x̃n, T x̃n, T x̃m))≤ F(µ(x̃n, x̃m),max{kSp(x̃n, x̃n, x̃m), Sp(x̃n, x̃n, x̃n), Sp(x̃m, x̃m, x̃m)})≤ F(1,max{kSp(x̃n, x̃n, x̃m), Sp(x̃n, x̃n, x̃n), Sp(x̃m, x̃m, x̃m)})

Therefore,

Sp(T x̃n, T x̃n, T x̃m) ≤ max{kSp(x̃n, x̃n, x̃m), Sp(x̃n, x̃n, x̃n), Sp(x̃m, x̃m, x̃m),Sp(x̃n, x̃n, x̃m) ≤ Un + Um + Sp(T x̃n, T x̃n, T x̃m)< Un + Um +max{kSp(x̃n, x̃n, x̃m), Sp(x̃n, x̃n, x̃n), Sp(x̃m, x̃m, x̃m)}.

Hence,

ρSp(α, µ) ≤ Sp(x̃n, x̃n, x̃m)

≤ max{23ϵ,2

3ϵ(1− k) + Sp(x̃n, x̃n, x̃n),

2

3ϵ(1− k) + Sp(x̃m, x̃m, x̃m)}

≤ max{23ϵ, ρSp(α, µ) + ϵ(1− k)} < ρSp(α, µ) + ϵ.

Thus,

limn,m

Sp(x̃n, x̃n, x̃m) = ρSp(α, µ).

Since (X,Sp) is complete, there exists a ∈ X such that,

Sp(a, a, a) = limnSp(a, a, x̃n) = lim

n,mSp(x̃n, x̃n, x̃m) = ρSp(α, µ).

Therefore, we have a ∈ XSp(α, µ) and thus XSp(α, µ) is nonempty. This impliesthat, ZSp(α, µ) is nonempty.

Now, let x0 ∈ ZSp(α, µ) be arbitrary. Then by the above argument we have

ρSp(α, µ) ≤ Sp(T x̃, T x̃, T x̃) ≤ Sp(x̃, x̃, T x̃) = Sp(x̃, x̃, x̃) = r0 = ρSp(α, µ).


Thus, T x̃ = x̃, Now, assume that T has two fixed points u, v ∈ ZSp(α, µ). Byour hypothesis, we know that α(u, v) ≥ 1, µ(u, v) ≤ 1. Thus,

h(1, Sp(u, u, v) ≤ h(α(u, v), Sp(Tu, Tu, Tv))≤ F(µ(u, v),max{kSp(u, u, v), Sp(u, u, u), Sp(v, v, v)})≤ F(1,max{kSp(u, u, v), Sp(u, u, u), Sp(v, v, v)}).

So we have,

Sp(u, u, v) ≤ max{kSp(u, u, v), Sp(u, u, u), Sp(v, v, v)}.

Now, if Sp(u, u, v) ≤ kSp(u, u, v) we deduce that Sp(u, u, v) = 0 and in thiscase u = v, or Sp(u, u, v) ≤ Sp(u, u, u) = Sp(v, v, v) and in this case by condi-tion (ii) of the definition of the partial S-metric space we obtain Sp(u, u, v) =Sp(u, u, u) = Sp(v, v, v) and hence by condition (i) of the same definition weconclude that u = v. Therefore, we obtain the uniqueness as desired.

As a consequence of the above result, the following corollary follows easily.

Corollary 2.14. Let (X,Sp) be a 0-complete partial S-metric space, k ∈ [0, 1)and consider the map T : X −→ X to be α-admissible and Rα-admissible,andthere exists x0 ∈ X such that α(x0, x0) ≥ 1, also for every x, y ∈ X we haveα(x, y)Sp(Tx, Tx, Ty) ≤ kSp(x, x, y). Then there exists x̃ ∈ X such that T x̃ = x̃.

Proof. Using the same technique and notation in the proof of Theorem 2.13, wededuce that Sp(xn, xn, xn) ≤ α(xn, xn)Sp(xn, xn, xn) ≤ knSp(x0, x0, x0). Thus,

r0 = Sp(x̃, x̃, x̃) = limnSp(x̃, x̃, xn) = limn,mSp(xn, xn, xm) = 0.

This implies that Sp(x̃, x̃, x̃) = 0. Since Sp(x̃, x̃, x̃) = Sp(x̃, x̃, T x̃) = 0, we havex̃ = T x̃ as required.

In closing, we change the contraction principle in Theorem 2.13, to showthat there exist a unique fixed point in the whole space X.

Theorem 2.15. Let (X,Sp) be a complete partial S-metric space, k ∈ [0, 1)and assume the there exists x0 ∈ X such that α(x0, x0) ≥ 1. Consider the mapT : X −→ X to be α−admissible and Rα−admissible. Assume that for everyx, y ∈ X we have

(2.2) α(x, y)Sp(Tx, Tx, Ty) ≤ max{kSp(x, x, y),Sp(x, x, x) + Sp(y, y, y)

2},

then there exists a unique u ∈ X such that Tu = u.

Proof. Note that, for every x, y ∈ X we have:

α(x, y)Sp(Tx, Tx, Ty) ≤ max{kSp(x, x, y),Sp(x, x, x) + Sp(y, y, y)

2}

≤ max{kSp(x, x, y), Sp(x, x, x), Sp(y, y, y)}.


Thus, all conditions of Theorem 2.13 are satisfied. Hence, there exists u ∈ Xsuch that Tu = u. Assume that there exist two fixed points u, v ∈ X for T suchthat α(u, v) ≥ 1. Hence,

Sp(u, u, v) = Sp(Tu, Tu, Tv) ≤ α(u, v)Sp(Tu, Tu, Tv)

≤ max{kSp(u, u, v),Sp(u, u, u) + Sp(v, v, v)

2}.

Thus, we either have Sp(u, u, v) ≤ kSp(u, u, v) which implies that Sp(u, u, v) = 0and hence u = v, or 0 = 2Sp(u, u, v)−Sp(u, u, u)−Sp(v, v, v) which also impliesthat u = v as desired.

Example 3. Let (X,Sp) be a partial S-metric space, where X = [0, 1] ∪ [2, 3]and the partial S-metric space Sp : X

3 −→ [0,+∞) is defined by

Sp(x, y, z) =

{∥max{x, y} − z∥, if {x, y, z} ∩ [2, 3] ̸= ∅|x− y − z|, if {x, y, z} ⊂ [0, 1].

Define the functions T : X −→ X and α : X × X −→ [0,∞) as follows Tx =x+ 1

2if 0 ≤ x ≤ 1, T2 = 1, and Tx = x+ 2

2if 2 < x ≤ 3,

α(x, y) =

{ex−y, if x ≥ y0, if x < y.

It is easy to see that T is α−admissible and Rα−admissible. Note that, we canalways pick our x, y and z such that max{x, y} > z. Also T is an increasingfunction. So, for every x ≥ y ∈ X we have:

Sp(Tx, Tx, Ty) ≤ α(x, y)Sp(Tx, Tx, Ty) ≤1

2Sp(x, x, y),

if x, y ∈ [0, 1], and

Sp(Tx, Tx, Ty) ≤ α(x, y)Sp(Tx, Tx, Ty) ≤Sp(x, x, x) + Sp(y, y, y)

2,

{x, y} ∩ [2, 3] ̸= ∅.One can verify that the function T in this example satisfies the conditions

of Theorem 2.15 and the unique fixed point will be 1.

References

[1] N. Mlaiki, A contraction principle in partial S-metric spaces, UniversalJournal of Mathematics and Mathematical Sciences, 5 (2014), 109-119.

[2] N. Mlaiki, A partially α-contractive principle, Advanced MathematicalStudies, 7 (2014), 121-126.


[3] S. Sedghi, N. Shobe and A. Aliouche, A generalization of fixed point theo-rems in S-metric spaces, Mat. Vesnik, 64 (2012), 258-266.

[4] S. G. Matthews, Partial metric topology, Ann. New York Acad. Sci., 728(1994), 183-197.

[5] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α − ψ−contactivetype mappings, Nonlinear Analysis, 75 (2012), 2154-2165.

[6] D. llic, V. Pavlovic, V. Rakcevic, Some new extentions of Banach’s con-traction principleto partial metric space, Applied Mathematics Letters, 24(2011), 1326-1330.

[7] T. Abdeljawad, E. Karapinar and K. Taş, Existence and uniqueness of acommon fixed point on partial metric spaces, Appl. Math. Lett., 24 (2011),1900-1904.

[8] T. Abdeljawad, E. Karapinar and K. Taş, A generalized contraction prin-ciple with control functions on partial metric spaces, Journal of Computerand Mathematics with Applications, 63 (2012), 716-719 .

[9] T. Abdeljawad, Fixed points for generalized weakly contractive mappings inpartial metric spaces, Math. Comput. Modelling, 54 (2011), 2923–2927.

[10] S. Oltra and O. Valero, Banach’s fixed point theorem for partial metricspaces, Rend. Istit. Mat. Univ. Trieste, 36 (2004), 17-26.

[11] O. Valero, On Banach fixed point theorems for partial metric spaces, Appl.Gen. Topol., 6 (2005), 229–240.

[12] T. Abdeljawad, Meir-Keeler alpha-contractive fixed and common fixed pointtheorems, Fixed point theory and applications, 19, DOI:10.1186/1687-1812-2013-19.

[13] H.P. Masiha, F. Sabetghadam, and N. Shahzad, fixed point theorems inpartial metric spaces with an application, Filomat, 27 (2013), 617-624.

[14] W. Shatanawi, A. Pitea, Some coupled fixed point theorems in quasi-partialmetric spaces, Fixed point theory and applications, 153 DOI: 10.1186/1687-1812-2013-153 Published: 2013.

[15] S.K. Malhotra, S. Shukla, R. Sen, A generalization of Banach contractionprinciple in ordered cone metric spaces, Journal of Advanced MathematicalStudies, 5 (2012), 59-67.

[16] Sana Hadj Amor, Tripled coincidence point theorems for weak Ï†-contractions in ordered partial metric spaces, Journal of Advanced Mathe-matical Studies, 6 (2013), 01-11.


[17] K. Abodayeh, A controlled contraction principle in partial S-metric spaces,Applied Mathematics and Information Sciences, 10 (2016), 1-5.

[18] Z. Kadelburg, H.K. Nashine and S. Radenoviä, Coupled fixed points inpartial metric spaces, Journal of advanced mathematical studies, 6 (2013),159-172.

[19] A. H. Ansari, Note on “α-admissible mappings and related fixed point the-orems”, The 2nd Regional Conference on Mathematics and Applications,PNU, September 2014, 373-376

[20] Arslan Hojat Ansari, Satish Shukla, Some fixed point theorems for orderedF -(F , h)-contraction and subcontractions in 0-f -orbitally complete partialmetric spaces, J. Adv. Math. Stud., 9(2016), 37-53

Accepted: 3.01.2017

ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 43–2020 (14–24) 14

Characterization of some linear groups by their conjugacy classsizes

Yanheng ChenKey Laboratory of Intelligent Information Processing and Control of ChongqingMunicipal Institutions of Higher EducationChongqing Three Gorges UniversityWanzhou, Chongqing, 404100, ChinaandSchool of Mathematics and StatisticsChongqing Three Gorges University, Wanzhou, Chongqing, 404100, P.R. China

Guiyun ChenSchool of Mathematics and Statistics, Southwest UniversityChongqing, 400715P.R. China

Yuming Feng∗

Key Laboratory of Intelligent Information Processing and Controlof Chongqing Municipal Institutions of Higher EducationChongqing Three Gorges UniversityWanzhou, Chongqing, 404100, P.R. ChinaandChongqing Engineering Research Center of Internet of Things and Intelligent ControlTechnologyChongqing Three Gorges UniversityWanzhou, Chongqing, 404100, [email protected]

B.O. OnasanyaKey Laboratory of Intelligent Information Processing and Control

of Chongqing Municipal Institutions of Higher Education

Chongqing Three Gorges University

Wanzhou, Chongqing, 404100

P.R. China

and

Department of Mathematics

University of Ibadan

Ibadan, 23402, Nigeria

Abstract. Let G be a group and denote by N(G) the set of conjugacy class sizesof G. In this paper, we proved that if Z(G) = 1 and N(G) = N(PGL(3, q)), whereq ∈ {2, 3, 4, 5, 7, 8, 9, 11}, then G ∼= PGL(3, q).Keywords: finite group, conjugacy class sizes, Thompson’s conjecture.

∗. Corresponding author

CHARACTERIZATION OF SOME LINEAR GROUPS BY THEIR CONJUGACY ... 15

1. Introduction

All groups considered in this paper are finite and simple groups are nonabelian.Let G be a group. For x ∈ G we denote by xG the conjugacy class of x, and by|xG| the size of xG. Then set N(G) = {|xG|

∣∣ x ∈ G}. It is a well-establishedtopic to investigate the relationship between the arithmetical properties of N(G)and the structural properties of group G. More recently, there have appeared anumber of papers addressing this research field. This paper is also a contributionalong this line, which is related to an open conjecture of John G. Thompson (ref.to [15, Problem 12.38]):

Thompson’s conjecture. If S is a simple group and G is a group satisfyingthat Z(G) = 1 and N(G) = N(S), then G ∼= S.

The prime graph of a group G is a simple graph whose vertices are the primedivisors of |G| and where two distinct primes p and q are joined by an edge if andonly if G contains an element of order pq. Using the prime graph of simple group,the second author proved that Thompson’s conjecture holds for all simple groupswith disconnected prime graph in 1994 (see [1], also ref. to [2, 3, 4]). For thesimple groups with connected prime graph, the conjecture has made considerableprogress in recent years. Several mathematicians had proved the conjecture istrue for the following simple groups: A10, A16, A22, U4(4), U4(5), An(q), Bn(q),Cn(q), Dn(q),

2Dn(q), and E7(q)(see [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]).A group M is said to be an almost simple related to S if and only if

S ≤ M ≤Aut(S) for some simple group S. Naturally, one can put forward thefollowing question: what are almost simple groups we can generalize Thomp-son’s conjecture to? Some authors have generalized the conjecture to almostsporadic simple groups except Aut(J2) and Aut(McL), symmetric groups Sn,where n = p, p+1, and p is an odd prime number, projective general linear groupsPGL(2, q), the automorphism groups of Suzuki-Ree groups (see[15, 16, 17, 18]).But they still used the second author’s method, which is only valid for the groupswith the disconnected prime graph.

In this paper, using Vasil’ev and Gorshkov’s methods, we generalized Thomp-son’s conjecture to projective general linear groups PGL(3, q), where q ∈{2, 3, 4, 5, 7, 8, 9, 11}. Note that PGL(3, 4) and PGL(3, 7) have the connectedprime graphs.

Our main result is the following theorem:

Main Theorem. Let G be a group with Z(G) = 1 and M one of groupsPGL(3, q), where q ∈ {2, 3, 4, 5, 7, 8, 9, 11}. If N(G) = N(M), then G ∼=M .

By [19], we get PGL(3, q) = L3(q).d, d = gcd(3, q − 1). Hence

M = PGL(3, q) =

{L3(q), q = 2, 3, 5, 8, 9, 11,

L3(q).3, q = 4, 7.

Since simple groups L3(2), L3(3), L3(5), L3(8), L3(9), L3(11) have discon-nected prime graphs, the second author in [1] has proved that Thompson’s

16 YANHENG CHEN, GUIYUN CHEN, YUMING FENG and B.O. ONASANYA

conjecture is right for these groups. Therefore, it is enough to prove Main The-orem for PGL(3, 4) and PGL(3, 7). We shall give the proofs on PGL(3, 4) inSection 3 and PGL(3, 7) in Section 4.

For convenience, we denote by π(n) the set of all primes dividing n wheren is a positive integer, and then nπ to denote π-part of n for π ⊆ π(n). Inaddition, for a group G, we also denote by π(G) = π(|G|), and Soc(G) the socleof G which is a subgroup generated by all minimal normal subgroups of G. Theother notation and terminologies in this paper are standard and the reader isreferred to [19] and [21] if necessary.

2. Preliminaries

First, we cite here some known results which are useful in the sequel.

Lemma 2.1. Let K be a normal subgroup of G and G = G/K. Then(a) If x is the image of an element x of G in the group G, then |xK |

∣∣|xG|and |xG|

∣∣|xG|.(b) If x ∈ G and (|x|, |K|) = 1, then CG(x) = CG(x)K/K.(c) If x, y ∈ G, (|x|, |y|) = 1, and xy = yx, then CG(xy) = CG(x)∩CG(y).

Lemma 2.2 ([5, Lemma 4]). Let G be a group with trivial center, p ∈ π(G) andp2 not divide n for any n ∈ N(G). Then a Sylow p-subgroup of G is elementaryabelian.

Lemma 2.3 ([9, Lemma 1.10]). Let a Sylow p-subgroup of G be of order p, xbe an element of order p, and |xG| be a number that is maximal with respect todivisibility in N(G). Then CG(x) is abelian.

Lemma 2.4 ([9, Lemma 1.9]). Let G be a group, and p and q be two numbersin π(G). If G satisfies the following conditions:

(a)N(G) contains no number divisible by p2 or q2;(b)N(G) contains no number except 1 co-prime to pq;(c)N(G) contains a number hq such that any n in N(G) not divisible by q

does not divide hq and N(G) contains no number divisible by hq and n;(d)N(G) contains a number hp such that any l in N(G) not divisible by p

does not divide hp and N(G) contains no number divisible by hp and l.Then Sylow p-subgroups and q-subgroups of G are cyclic groups of prime order.In addition, G has no element of order pq.

Lemma 2.5 ([9, Lemma 1.12]). Let G be a group, K the soluble radical ofG, and G/K = S a simple group. Suppose that there exists a prime p suchthat p ∈ π(G) \ π(K). Assume that an element g of order p of G satisfies thefollowing conditions:

(a) |gG| = |gS |, where g is the image of an element g in the group S ;(b) the number |gG| is maximal with respect to divisibility in N(G);


(c) the subgroup CG(g) is abelian.

Then K ≤ Z(G).Let M be one of PGL(3, 4) and PGL(3, 7). Information on the set N(M)

and the order of M given in the next two lemmas is obtained via [19] or GAP[22].

Lemma 2.6. Let M ∼= PGL(3, 4). Then:(1) |M | = 26 · 33 · 5 · 7;(2) N(M) = {n1 = 1, n2 = 32 · 5 · 7, n3 = 24 · 3 · 7, n4 = 26 · 3 · 5, n5 =

26 · 5 · 7, n6 = 26 · 32 · 5,n7 = 2

2 · 33 · 5 · 7, n8 = 26 · 32 · 7, n9 = 24 · 32 · 5 · 7}.Especially,

(3) N(M) contains no number other than n1, n4 and n6 not divisible by 7;


(5) For any n ∈ N(M) and p ∈ {5, 7}, it follows that p2 ̸ |n;(6) |xM | = n1, x ∈M if and only if x = 1.

Lemma 2.7. Let M ∼= PGL(3, 7). Then:(1) |M | = 25 · 33 · 73 · 19;(2) N(M) = {n1 = 1, n2 = 24 · 32 · 19, n3 = 3 · 72 · 19, n4 = 25 · 3 · 73, n5 =

23 · 73 · 19,n6 = 2

5 ·32 ·73, n7 = 25 ·33 ·7 ·19, n8 = 2 ·32 ·73 ·19, n9 = 24 ·32 ·72 ·19, n10 =23 · 3 · 73 · 19}.In particular,


(4) N(M) contains no number divided by 192;

(5) |xM | = n1, x ∈M if and only if x = 1.

Lemma 2.8. If M is one of PGL(3, 4) and PGL(3, 7), and G is a group withZ(G) = 1 and N(G) = N(M), then |M |

∣∣|G| and π(G) = π(M).Proof. Since the number in N(G) divides |G|, under the hypothesis we see that|M |

∣∣|G| by Lemma 2.6 and Lemma 2.7. π(M) = π(G) is the result of Lemma1.2.1 in [1] or Lemma 3 in [5].

Lemma 2.9. Let S be a simple group.

(i) If π(S) ⊆ {2, 3, 5, 7}, then S is isomorphic to one of simple groups ofTable 1.

(ii) If π(S) ⊆ {2, 3, 7, 19}, then S is isomorphic to one of simple groupsof Table 2.

Proof. This is an immediate consequence of Theorem 2 in [23].

For convenience, we list all the cases of S in Lemma 2.9 as well as the ordersof S, the orders of the outer automorphism of S in Table 1 and Table 2.

18 YANHENG CHEN, GUIYUN CHEN, YUMING FENG and B.O. ONASANYA

Table 1. Non-abeian simple groups S with π(S) ⊆ {2, 3, 5, 7}

S Order of S |Out(S)| S Order of S |Out(S)|A5 2

2 · 3 · 5 2 A9 26 · 34 · 5 · 7 2L2(7) 2

3 · 3 · 7 2 J2 27 · 33 · 52 · 7 2A6 2

3 · 32 · 5 22 U3(5) 24 · 32 · 53 · 7 |S3|L2(8) 2

3 · 32 · 7 3 S6(2) 29 · 34 · 5 · 7 1A7 2

3 · 32 · 5 · 7 2 U4(3) 27 · 36 · 5 · 7 |D8|U3(3) 2

5 · 33 · 7 2 S4(7) 28 · 32 · 52 · 74 2A8 2

6 · 32 · 5 · 7 2 A10 27 · 34 · 52 · 7 2L3(4) 2

6 · 32 · 5 · 7 |D12| O+8 (2) 212 · 35 · 52 · 7 |S3|U4(2) 2

6 · 34 · 5 2 L2(49) 24 · 3 · 52 · 72 22

Table 2. Non-abeian simple groups S with π(S) ⊆ {2, 3, 7, 19}

S Order of S |Out(S)| S Order of S |Out(S)|L2(7) 2

3 · 3 · 7 2 L2(8) 23 · 32 · 7 3U3(3) 2

5 · 33 · 7 2 L3(7) 25 · 32 · 73 · 19 |S3|U3(8) 2

7 · 34 · 7 · 19 |3× S3|

3. Proof of the main theorem for PGL(3, 4)

Theorem 3.1. Let G be a group with trivial center. If N(G) = N(PGL(3, 4)),then G ∼= PGL(3, 4).

Proof. We divide the proof of this theorem into six steps.

Step 1. Sylow 5-subgroups and Sylow 7-subgroups of G are cyclic groups ofprime order and there are no elements of order 35 in G.

In view of N(G) = N(PGL(3, 4)) and Lemma 2.8, we can choose p = 5 andq = 7, and take h5 = n6 and h7 = n8 such that G satisfies the hypotheses ofLemma 2.4. Hence Sylow 5-subgroups and Sylow 7-subgroups of G are cyclicgroups of prime order and there are no elements of order 5 · 7 in G.

Step 2. Let g, h ∈ G be elements of orders 5 and 7, respectively. Then |gG| =n8 = 2

6 · 32 · 7 and |hG| = n6 = 26 · 32 · 5, and CG(g) and CG(h) are abelian.Since the Sylow 5-subgroup of G is order of 5 by Step 1, one has that 5 ̸

∣∣ |xG|for any 1 ̸= x ∈ CG(g). Hence |xG| = n3 or n8 by (4) and (6) of Lemma 2.6.Assume that |gG| = n3 = 24 · 3 · 7. Let H be a Sylow 3-subgroup of CG(g).Then H is a nontrivial group of order |G|3/3 by Lemma 2.8. It follows thatZ(H) ̸= 1 and let 1 ̸= y ∈ Z(H). Then H ≤ CG(y), and so |yG|3 ≤ 3.Thus |yG| = n3. Since H ≤ CG(gy), we have that |(gy)G| = n3. In viewof CG(gy) = CG(g)

∩CG(y), we see that CG(g) = CG(y). The group CG(y)


contains an element w from the center of a Sylow 3-subgroup of G, then |wG|is not divisible by 3, and so |wG| = n5 = 26 · 5 · 7 by (2) of Lemma 2.6. Thusw ̸∈ CG(g), a contradiction. It follows that |gG| = n8 = 26 · 32 · 7. Since n8 ismaximal with respect to divisibility in N(G), Lemma 2.3 implies that the groupCG(g) is abelian.

In a similar way, we can show that |hG| = n6 = 26 · 32 · 5 and CG(h) isabelian.

In the following discussion, we assume that K is the soluble radical of agroup G, and G = G/K.

Step 3. G is non-soluble and has a unique composition factor S such that5 · 7

∣∣|S| and S E G ≤Aut(S). Moreover, S may be isomorphic to one of thefollowing groups:

A7, A8, A9, L4(3), S6(2), U4(3).

Assume that 5∣∣|K|. Then K/O{5,7}′(K) has a normal subgroup T of order 5.

Hence an element of G/O{5,7}′(K) of order 7 can act trivially on T , which impliesthat G/O{5,7}′(K) contains an element of order 35, so does G, contradicting withStep 1. Thus 5 does not divide |K| and similarly we can prove that 7 does notdivide |K|, and so G is not soluble.

Let L = S1 × S2 × · · · × Sk be the socle of G, where S1, S2, . . . , and Sk aresimple groups. Let g be an element of order 5 of G and suppose that 5 ̸∈ π(L).Then g is of order 5 in G and induces a nontrivial outer automorphism of thegroup L. Suppose that there exists i such that Sgi ̸= Si. Without loss ofgenerality, we assume that i = 1. Let H = ⟨s |s = s1sg1s

g2

1 sg3

1 sg4

1 , s1 ∈ S1⟩.Then H lies in the centralizer of the element g and is isomorphic to S1, butthe centralizer of g is abelian by Step 2, a contradiction. Hence g induces anontrivial outer automorphism of the group Si such that 5

∣∣|Out(Si)|. In view ofπ(Si) ⊆ π(G) = {2, 3, 5, 7} and by Table 1, the prime divisors of |Out(Si)| areless than 5, a contradiction. Therefore 5

∣∣|L| and sim

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Italian Journal of Pure and Applied Mathematicsijpam.uniud.it/online_issue/IJPAM_no-43-2020.pdf · [email protected] Giovanni Falcone Dipartimento di Metodi e Modelli Matematici