Dihedral p-critical elements in finite simple groupsgrgr/SlidesEdmontonMargolis.pdf · Dihedral p-critical elements in ﬁnite simple groups Leo Margolis University of Stuttgart Groups,

Dihedral p-critical elements in finite simplegroups

Leo Margolis

University of Stuttgart

Groups, Rings and Group Rings, 2011

Leo Margolis Dihedral p-critical elements in finite simple groups

Free subgroups in U(ZG)

Assume G is a finite group.When does U(ZG) contain a free non-abelian subgroup?Obviously not, if U(ZG) is abelian or finite.

Theorem (Hartley-Pickel 1980), non-constructive

If U(ZG) is not finite nor abelian, then U(ZG) contains annon-abelian free subgroup.


Free subgroups in U(ZG)

Assume G is a finite group.When does U(ZG) contain a free non-abelian subgroup?Obviously not, if U(ZG) is abelian or finite.

Theorem (Hartley-Pickel 1980), non-constructive

If U(ZG) is not finite nor abelian, then U(ZG) contains annon-abelian free subgroup.


Construction of free subgroups in U(ZG)

Theorem (Marciniak-Sehgal 1997)

Let u be a non-trivial bicyclic unit. Then 〈u,u∗〉 is non-abelianfree.

Theorem (Ferraz 2003)

Construction of a free n.-a. subgroup in U(Z(Q8 × Cp)) with pan odd prime using Bass-cyclic units.

→ Construction of free n.-a. subgroup in all possible cases.


The conjecture of Á del Río and J. Z. Gonçalves

Conjecture (del Río, Gonçalves)

Let u be a Bass-cyclic unit based on an element a ∈ G of primeorder at least 5 and suppose u has infinite order modulo thecenter of U(ZG). Then there is a Bass-cyclic or bicyclic unit vand an integer n such that 〈un, vn〉 is non-abelian free.(Proof given over solvable groups)

LemmaLet u be a Bass-cyclic unit based on an element a ∈ G of primeorder at least 5. TFAE

u has infinite order modulo the center of U(ZG).u has infinite order and the conjugacy class of a in G doesnot lie in {a,a−1}.


The conjecture of Á del Río and J. Z. Gonçalves

Conjecture (del Río, Gonçalves)

Let u be a Bass-cyclic unit based on an element a ∈ G of primeorder at least 5 and suppose u has infinite order modulo thecenter of U(ZG). Then there is a Bass-cyclic or bicyclic unit vand an integer n such that 〈un, vn〉 is non-abelian free.(Proof given over solvable groups)

LemmaLet u be a Bass-cyclic unit based on an element a ∈ G of primeorder at least 5. TFAE

u has infinite order modulo the center of U(ZG).u has infinite order and the conjugacy class of a in G doesnot lie in {a,a−1}.


Dihedral p-critical elements

For a ∈ G define DG(a) = {g ∈ G | ag ∈ {a,a−1}} to be thedihedralizer of a in G. An element a in G of prime order p atleast 5 is called dihedral p-critical if it satisfies

DG(a) 6= G,DU(a) = U for U � G, a ∈ U,DG/N(a) = G/N for N E G.

LemmaLet u, a Bass-cyclic unit based on a ∈ G, be a minimalcounterexample to the conjecture (minimal in respect to |G|).Then a is dihedral p-critical.


Dihedral p-critical elements

For a ∈ G define DG(a) = {g ∈ G | ag ∈ {a,a−1}} to be thedihedralizer of a in G. An element a in G of prime order p atleast 5 is called dihedral p-critical if it satisfies

DG(a) 6= G,DU(a) = U for U � G, a ∈ U,DG/N(a) = G/N for N E G.

LemmaLet u, a Bass-cyclic unit based on a ∈ G, be a minimalcounterexample to the conjecture (minimal in respect to |G|).Then a is dihedral p-critical.


Groups containing dihedral p-critical elements

PropositonLet a be a dihedral p-critical element of a finite group G. Thenone of the following hold:(1) G = 〈a〉p o 〈b〉q for q = 4 or an odd prime such that ab = ai ,

where q equals the order of i modulo p.(2) G = (〈a〉p × 〈z〉p)o 〈b〉p with z ∈ Z (G) und ab = za.

(3) G = 〈b〉pn o 〈a〉p with n ≥ 2 und ba = b1+pn−1.

(4) G = (〈a〉p × 〈z〉p)o 〈b〉2 with z ∈ Z (G) and ab = za−1.(5) G = Ao 〈b〉q , where A is an elementary-abelian non-cyclic

p-group containing a, q a prime with q 6= p and the action of 〈b〉on A is faithful and irreducible.

(6) G = (〈a〉p × 〈a1〉p)o 〈b〉4 with ab = a1 and ab1 = a−1.

(7) G = B o 〈a〉p, where B is an elementary-abelian q-group and aacts faithful and irreducible on B.

(8) G is simple.


An example

Consider G = An. Which dihedral p-critical elements a does Gcontain?

For n = 5 the 5-elements are dihedral p-critical.If for n ≥ 6 an element fixes a number in {1, ...,n}, it lies ina subgroup U ∼= An−1 and is therefore not dihedralp-critical. Assume n is not prime and

a = (a11, ...,ap

1)...(a1k , ...,ap

k ), g = (a11,a2

1)(a31,a4

1),

then ag /∈ {a,a−1}, but 〈a,g〉 6= G.So n = p, but then a is conjugate to some ak and ak 6= a±1

and a is not dihedral p-critical.→ Only A5 contains dihedral p-critical elements.


An example



a = (a11, ...,ap

1)...(a1k , ...,ap

k ), g = (a11,a2

1)(a31,a4

1),




An example



a = (a11, ...,ap

1)...(a1k , ...,ap

k ), g = (a11,a2

1)(a31,a4

1),




An example



a = (a11, ...,ap

1)...(a1k , ...,ap

k ), g = (a11,a2

1)(a31,a4

1),




Dihedral p-critical elements in finite simple groups

Theorem (R. Guralnick 2010 using CFSG)

Let G be a finite non-abelian simple group containing dihedralp-critical elements. Then G is isomorphic to a PSL(2,q).

→ This results gives hope to prove it without the CFSG.


First results

From here on let G be a finite n.-a. simple group containing adihedral p-critical element a. Write Cl(a) for the conjugacyclass of a in G. a ∼ b means b ∈ Cl(a).Elementary results:

For g /∈ DG(a) we have 〈a,g〉 = G and DG(a) is the onlymaximal subgroup containing a.Every conjugate of a is dihedral p-critical.For b ∼ a and b /∈ {a,a−1} we have 〈a,b〉 = G (usessimplicity of G).An element g ∈ G is determined by ag and bg . EspeciallyCG(a) ∩ CG(b) = 1.a ∼ a−1 for otherwise CG(a) would be aFrobenius-complement.DG(a)∩DG(b) is an elementary abelian 2-group of order atmost 4 and DG(a) ∩ DG(b) 6= 1 for some b.


First results




First results




First results




First results

Lemma|Cl(a)| ≡ 2 mod |U| for every U ≤ CG(a). Especially if q is anodd prime dividing |CG(a)|, then CG(a) contains an q-Sylowsubgroup of G and if 4||CG(a)|, then DG(a) contains a 2-Sylowsubgroup of G.


The case |CG(a)| even

From now on let |CG(a)| be even and let i be an involution in Gsuch that ai = a−1.

Using transfer theory one finds b ∼ a with b /∈ {a,a−1}such that DG(a) ∩ DG(b) ∼= C2 × C2.This gives CDG(a)(i)

∼= C2 × C2 and even CP(i) ∼= C2 × C2for some 2-Sylow subgroup P of G containing i .This only happens, if P is a dihedral or semidihedral group(which means P = 〈x , i | x2n−1

= i2 = 1, x i = x−1+2n−2〉).






= i2 = 1, x i = x−1+2n−2〉).






= i2 = 1, x i = x−1+2n−2〉).



By big works of Gorenstein and Walter and from Alperin,Brauer and Gorenstein this 2-Sylow subgroups occur onlyin the following finite simple groups:

PSL(2,q) for q ≡ 1 mod 2 or A7 (in the dihedral case),PSL(3,q) for q ≡ −1 mod 4 or PSU(3,q) for q ≡ 1 mod 4or M11 (in the semidihedral case).

The local structure also gives NG(P) = P for a 2-Sylowsubgroup P. Of the groups listed above only PSL(2,q) and5 other groups satisfy this condition.Use GAP to kill the rest

→ In this case Guralnicks result can be observed without theCFSG.I think also in the odd case, but I don’t know how exactly.




















Thank you for your attention and enjoy theConference dinner!


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Dihedral p-critical elements in finite simple groupsgrgr/SlidesEdmontonMargolis.pdf · Dihedral p-critical elements in ﬁnite simple groups Leo Margolis University of Stuttgart Groups,