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On the Chermak-Delgado Lattices of Split Metacyclic p-Groups Research by
Brianne Power,Erin Brush, and Kendra Johnson-TeschSupervised by Jill Dietz at St. Olaf
College
Chermak and Delgado (1989) were interested in finding families of characteristic subgroups. They introduced a measure that was later deemed the “Chermak-Delgado” measure. The subgroups with maximalChermak-Delgado measure form a lattice.
Not many Chermak-Delgado lattices have been unearthed due to their complexity. These lattices give a visual representation of deep structural properties of finite p-groups and their subgroups.
Background
Andrew ChermakKansas State University
Alberto DelgadoIllinois State University
Useful Definitions
Center of G: The set of elements in a group G that commute with every element in G
Z(G) = { z ϵ G | zg = gz for all g ϵ G }
Centralizer of S: The set of elements in G that commute with all of the elements in a subset S of G
CG(S) = { c ϵ G | sc = cs for all s ϵ S }
The Chermak-Delgado Measure
The Chermak-Delgado measure of a subgroup H is G is mG(H) = |H| |CG(H)|.
We write m*(G) to denote the largest possible Chermak-Delgado measure of the subgroups of G.
The Chermak-Delgado Lattice
The Chermak-Delgado lattice of a finite group G is a lattice comprised of subgroups of G with the largest possible Chermak-Delgado measure. For the finite group G, we writeCD(G) for the Chermak-Delgado lattice of G.
Example 1: The abelian group Z6
G = Z6 = {0, 1, 2, 3, 4, 5} H1 = {0, 2, 4} H2 = {0, 3} H3 = {0}
mG(G) = |G| |CG(G)| = |G| |Z(G)| = |G|2 = 62 = 36 mG(H1) = |H1| |CG(H1)| = |H1| |G| = 3.6 = 18mG(H2) = |H2| |CG(H2)| = |H2| |G| = 2.6 = 12mG(H3) = |H3| |CG(H3)| = |H3| |G| = 1.6 = 6
← m*(G)
Generalization: Abelian Groups
Let A be an abelian group.
m*(A) = mA(A)
= |A| |CA(A)|
= |A| |Z(A)|
= |A|2
Example 2: Dihedral group D8
Presentation of D8: < x, y | x4 = 1 = y2, yxy-1 = x3 >
(the dihedral group of order 8)
x y
Rotation
Reflection
G = D8
H1 = <x2,y>H2 = <x>H3 = <x2,xy>H4 = <y>H5 = <x2y>H6 = <x2>H7 = <xy>H8 = <x3y>e
Example 2: Dihedral group D8
mG(G) = |G| |Z(G)| = |G| |H6| = 8.2 = 16
mG(H1 ) = 16mG(H2 ) = 16mG(H3 ) = 16mG(H4 ) = 8mG(H5 ) = 8mG(H6 ) = 16mG(H7 ) = 8mG(H8 ) = 8mG(e) = 8
m*(G)=16
Metacyclic p-Groups● G is metacyclic if it has a cyclic normal subgroup
N such that G/N is also cyclic
● Metacylic groups are generated by two elements x and y where:o x generates No yN generates G/N
● A metacylic p-group has pk elements (p a prime)
P(p,m): A family of metacyclic p-groups
P(p,m) = < x, y | xp^m = 1 = yp, yxy-1 = x1+p^(m-1) >
Note: D8=P(2,2)
Observe: |P| = pm+1, Z(P) = <xp>, |Z(P)| = pm-1, mP(P)=p2m
Theorems: m*(P) = p2m CD(P) contains p+3 subgroups
P(p,m): How to Prove
1. Gather information about all subgroups of P
2. Find centralizers using known relations
3. Apply properties of p-groups and normal subgroups
Generalize to other metacyclic groups
P(p,m) = < x, y | xp^m = 1 = yp, yxy-1 = x1+p^(m-1) >P(p,m,1,1) = < x, y | xp^m = 1 = yp^1, yxy-1 = x1+p^(m-1) >P(p,m,n,r) = < x, y | xp^m = 1 = yp^n, yxy-1 = x1+p^(m-r) >
A Broader Family of Metacyclics
P(p,m,n,r) = < x, y | xp^m = 1 = yp^n, yxy-1 = x1+p^(m-r) >
where m > 2, n > 1, and 1 < r < min{m-1, n}
Observations: |P| = pm+n and Z(P) = <xp^r, yp^r>
Theorem: mP(P) = p2(m+n-r)
mP(P) ≟ m*(P)
A Broader Family of Metacyclics
Theorem:m*(P) = p2(m+n-r) = mP(P)
This means that the lattice is a sublattice ofCD(P)!
P(p,m,n,r): How we found the lattice
1. Used examples and tested out patterns
2. Applied properties of p-groups and normal subgroups
3. External research confirmed that the measure of these groups is the maximal measure of P
Current Research● Confirmation that our lattice is a
sublattice of CD(P)
● What else is in CD(P)?
● What does the lattice of all subgroups of P look like?
● Investigate other measures identified by Chermak and Delgado
Research Sources● L. An, J. Brennan, H. Qu, and E. Wilcox, Chermak-Delgado lattice extension theorems, submitted, 2013.
http://arxiv.org/pdf/1307.0353v1.pdf● Y. Berkovich, Maximal abelian and minimal nonabelian subgroups of some finite two-generator p-groups especially
metacyclic, Israel J. Math. 194 (2013), 831-869.● J.N.S. Bidwell and M.J. Curran, The automorphism group of a split metacyclic p-group, Math. Proc. R. Ir. Acad. 110A
(2010), no. 1, 57-71.● B. Brewster, P. Hauck, and E. Wilcox, Groups whose Chermak-Delgado lattice is a chain, submitted, 2013.
http://arxiv.org/pdf/1305.2327v1.pdf● B. Brewster and E. Wilcox, Some groups with computable Chermak-Delgado lattices, Bull. Aus. Math. Soc. {86 (2012), 29-
40.● A. Chermak and A. Delgado, A measuring argument for finite groups, Proc. AMS 107 (1989), no. 4, 907-914.● G. Glauberman, Centrally large subgroups of finite p-groups, J. Algebra 300 (2006), no. 2, 480-508.● L. Héthelyi and B, Külshammer, Characters, conjugacy classes and centrally large subgroups of p-groups of small rank, J.
Algebra 340 (2011), 199-210.● I. M. Isaacs, Finite Group Theory, American Mathematical Society, 2008.● King, Presentations of Metacyclic Groups, Bull. Aus. Math. Soc. 8 (1973), 103-131.● W.K. Nicholson, Introduction to Abstract Algebra, 4th Edition, Wiley, 2012.● M. Schulte, Automorphisms of metacyclic p-groups with cyclic maximal subgroups, Rose-Hulman Undergraduate
Research Journal 2 (2001), no. 2.● M. Suzuki, Group Theory II, Springer-Verlag, 1986.
Image Sourceshttp://www.math.ksu.edu/people/personnel_detail?person_id=1326
https://faculty.sharepoint.illinoisstate.edu/aldelg2/Pages/default.aspx
http://www.quickmeme.com/Bad-Joke-Eel/page/565/
http://fergalsresearch.weebly.com/subgroup-lattices.html