BOUNDEDLY GENERATED SUBGROUPS OF FINITE GROUPS

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1 BOUNDEDLY GENERATED SUBGROUPS OF FINITE GROUPS Andrea Lucchini Università di Padova, Italy 2nd Biennial International Group Theory Conference Istanbul, February 4-8, 2013

2 AN (EASY?) QUESTION Assume that G is a finite group and let π(g) be the set of prime divisors of the order of G. Does there exist a 2-generated subgroup H of G with π(h) = π(g)?

3 AN (EASY?) QUESTION Assume that G is a finite group and let π(g) be the set of prime divisors of the order of G. Does there exist a 2-generated subgroup H of G with π(h) = π(g)? YES! and a stronger result can be proved:

4 AN (EASY?) QUESTION Assume that G is a finite group and let π(g) be the set of prime divisors of the order of G. Does there exist a 2-generated subgroup H of G with π(h) = π(g)? YES! and a stronger result can be proved: THEOREM (AL, M. MORIGI, P. SHUMYATSKY (2011)) Let C(G) be the set of isomorphism classes of composition factors of G, then there exists a 2-generated subgroup H of G such that C(H) = C(G).

5 HOW A RESULT LIKE THIS CAN BE PROVED? Let L be a primitive monolithic group (L has a unique minimal normal subgroup A, and if A is abelian then A has a complement in G). The crown-based power of L of size k is the subgroup L k of L k defined by: L k = {(l 1,..., l k ) L k l 1... l k mod A}. Let d(g) denotes the minimal number of generators of the group G. THEOREM (F. DALLA VOLTA, AL (1998)) Let m be a natural number and let G be a finite group such that d(g/n) m for every non-trivial normal subgroup N, but d(g) > m. Then there exists a primitive monolithic group L such that G = L k for some k.

6 Let H G, minimal with the property that C(H) = C(G).

7 Let H G, minimal with the property that C(H) = C(G). There exists N H such that: d(h/n) = d(h) but d(h/m) < d(h) whenever N < M H.

8 Let H G, minimal with the property that C(H) = C(G). There exists N H such that: d(h/n) = d(h) but d(h/m) < d(h) whenever N < M H. There exist L and k with H/N = L k.

9 Let H G, minimal with the property that C(H) = C(G). There exists N H such that: d(h/n) = d(h) but d(h/m) < d(h) whenever N < M H. There exist L and k with H/N = L k. There exists N H H with H /N = L. Since C(H) = C(H ), it must be H = H, hence k = 1.

10 Let H G, minimal with the property that C(H) = C(G). There exists N H such that: d(h/n) = d(h) but d(h/m) < d(h) whenever N < M H. There exist L and k with H/N = L k. There exists N H H with H /N = L. Since C(H) = C(H ), it must be H = H, hence k = 1. THEOREM (AL, F. MENEGAZZO (1997)) If L is a primitive monolithic group, then d(l) = max(2, d(l/ soc L)).

11 Let H G, minimal with the property that C(H) = C(G). There exists N H such that: d(h/n) = d(h) but d(h/m) < d(h) whenever N < M H. There exist L and k with H/N = L k. There exists N H H with H /N = L. Since C(H) = C(H ), it must be H = H, hence k = 1. THEOREM (AL, F. MENEGAZZO (1997)) If L is a primitive monolithic group, then d(l) = max(2, d(l/ soc L)). Let M/N = soc H/N : d(h) = d(h/n) = max{2, d(h/m)} = 2.

12 The previous proof suggests a strategy to tackle other similar (more difficult) questions: QUESTION Let i(g) be a group invariant. Does there exist d N such that every finite group G contains a d-generated subgroup H with i(h) = i(g)? STRATEGY Let H G minimal such that i(h) = i(g) and let d = d(h). There is N H s.t. d(h/n) = d, d(h/m) < d N < M H. H/N = L k for suitable L and k. Can we prove that k t, where t depends on the invariant we are considering but not on H? If the previous question has a positive answer and we can prove that there exists an absolute constant τ such that d(l t ) max(d(l), τ) for any choice of L, then we can conclude that d(h) τ.

13 d(l k ) f (k) Let L be a primitive monolithic group, A = soc L. Assume d(l) d. ASSUME THAT A IS ABELIAN Let q = End L/A (A), q r = A, q s = H 1 (L/A, A), θ = 0 or 1 according to whether A is a trivial L/A-module or not. Then d(l k ) d k r(d θ) s. Notice that s < r, as a consequence of a result of Aschbacher and Guralnick, indeed we have: H 1 (L/A, A) < A. In particular d(l k ) max(d(l), (k + s)/r + θ) max(d(l), k + 1).

14 d(l k ) f (k) Let L be a primitive monolithic group, A = soc L. Assume d(l) d. ASSUME THAT A IS NON ABELIAN Let P L,A (d) be the conditional probability that d randomly chosen elements of L generate L given that they generate L modulo A. Then d(l k ) d k P L,A(d) A d C Aut(A) (L/A). A = S n where n is a positive integer and S is a nonabelian simple group. It is not difficult to prove that C Aut(A) (L/A) n S n 1 Aut(S). We need an estimation of P L,A (d).

15 d(l k ) f (k) Let L be a primitive monolithic group with non-abelian socle A = S n. THEOREM (E. DETOMI, C. RONEY-DOUGAL, AL (2013)) If d d(l) then there exists a 2-generated almost simple group Y with socle S such that P L,A (d) P Y,S (2). THEOREM (N. MENEZES, M. QUICK, C. RONEY-DOUGAL (2012)) If Y is a 2-generated almost simple group with non abelian socle S, then P Y,S (2) 53/90 with equality if and only if Y = Alt(6), Sym(6). It follows: if d(l) d, then P L,A (d) 53/90. Moreover P L,A (d) 1 as A.

16 PRIME GRAPH Denote by Γ(G) the prime graph of a finite group G. This is the graph whose set of vertices is π(g) and p, q π(g), with p q, are connected by an edge if and only if G has an element of order pq.

17 PRIME GRAPH Denote by Γ(G) the prime graph of a finite group G. This is the graph whose set of vertices is π(g) and p, q π(g), with p q, are connected by an edge if and only if G has an element of order pq. THEOREM (M. MORIGI, P. SHUMYATSKY, AL (2011)) Let G be a finite group. Then there exists a 3-generated subgroup H of G such that Γ(H) = Γ(G).

18 PRIME GRAPH Denote by Γ(G) the prime graph of a finite group G. This is the graph whose set of vertices is π(g) and p, q π(g), with p q, are connected by an edge if and only if G has an element of order pq. THEOREM (M. MORIGI, P. SHUMYATSKY, AL (2011)) Let G be a finite group. Then there exists a 3-generated subgroup H of G such that Γ(H) = Γ(G). The previous theorem cannon be improved.

19 V 1 = F 5 F 5, V 2 = F 7 F 7. ( ) ( ) Sym(3) = α 1 :=, β := GL(V ). ( ) ( ) Sym(3) = α 2 :=, β := GL(V ). (Sym(3)) 2 = K = (β1, 1), (1, β 2 ), (α 1, α 2 ) GL(V 1 ) GL(V 2 ). G := (V 1 V 2 ) K. d(g) = 3, but Γ(H) Γ(G) for all H < G.

20 V 1 = F 5 F 5, V 2 = F 7 F 7. ( ) ( ) Sym(3) = α 1 :=, β := GL(V ). ( ) ( ) Sym(3) = α 2 :=, β := GL(V ). (Sym(3)) 2 = K = (β1, 1), (1, β 2 ), (α 1, α 2 ) GL(V 1 ) GL(V 2 ). G := (V 1 V 2 ) K. d(g) = 3, but Γ(H) Γ(G) for all H < G g g V β 2 21 g g V β 1

21 SPECTRUM The spectrum of a group G is the set of orders of the elements of G.

22 SPECTRUM The spectrum of a group G is the set of orders of the elements of G. For every positive integer d there exists a group G such that d(g) = d and no (d 1)-generated subgroup of G has the same spectrum.

23 SPECTRUM The spectrum of a group G is the set of orders of the elements of G. For every positive integer d there exists a group G such that d(g) = d and no (d 1)-generated subgroup of G has the same spectrum. X = {p 1,..., p d } a set of d different odd prime numbers D i = b i, a i b p i i = ai 2 = 1, b a i i = b 1 i = D 2pi, G = d i=1 D i if d 2, then d(g) = d the spectrum of G is the set of the proper divisors of m = 2p 1 p d the elements of order m/p i are of the form (b r 1 1,..., b r i 1 i 1, a ib r i i, b r i+1 i+1,..., br d d ).

24 COMPLEX IRREDUCIBLE CHARACTER DEGREES Let π cd (G) be the set of the prime divisors of the complex irreducible character degrees of G.

25 COMPLEX IRREDUCIBLE CHARACTER DEGREES Let π cd (G) be the set of the prime divisors of the complex irreducible character degrees of G. THEOREM Let G be a finite group. There exists a 3-generated subgroup H of G such that π cd (H) = π cd (G).

26 COMPLEX IRREDUCIBLE CHARACTER DEGREES Let π cd (G) be the set of the prime divisors of the complex irreducible character degrees of G. THEOREM Let G be a finite group. There exists a 3-generated subgroup H of G such that π cd (H) = π cd (G). The previous theorem cannot be improved. The non abelian group P of order 27 and exponent 3 admits an automorphism α of order 2, acting on P/ Frat(P) as the inverting automorphism. Let G = P α : d(g) = 3 and π cd (G) = {2, 3}. Let H < G : π cd (H) = {3} if H P, π cd (H) = {2} otherwise.

27 CONJUGACY CLASS SIZES Let π cs (G) be the set of the prime divisors of the conjugacy class sizes of G.

28 CONJUGACY CLASS SIZES Let π cs (G) be the set of the prime divisors of the conjugacy class sizes of G. For every positive integer d there exists a finite group G such that if H G and π cs (H) = π cs (G) then d(h) d.

29 CONJUGACY CLASS SIZES Let π cs (G) be the set of the prime divisors of the conjugacy class sizes of G. For every positive integer d there exists a finite group G such that if H G and π cs (H) = π cs (G) then d(h) d. To each σ = {i, j} {1,..., t}, with i j, we associate a different prime p σ. Let A σ be a cyclic group of order p σ and A = σ A σ. For 1 i t, let C i = x i be a cyclic group of order 2 and let C = i C i. We define an action of C on A: x i centralizes A σ if i / σ, x i acts of A σ as the inverting automorphism otherwise. Let G = A C; π cs (G) = π(g) = {2} {p σ σ}. H G and π cs (H) = π cs (G) = d(h) log 2 t.

30 Denote by Γ cd (G) the graph whose set of vertices is π cd (G) and p, q π(g), with p q, are connected by an edge if and only if p q divides the degree of an irreducible complex character of G.

31 Denote by Γ cd (G) the graph whose set of vertices is π cd (G) and p, q π(g), with p q, are connected by an edge if and only if p q divides the degree of an irreducible complex character of G. OPEN QUESTION Does there exist a positive integer c with the property that every finite group G contains a c-generated subgroup H with Γ cd (H) = Γ cd (G)?

32 EXPONENT THEOREM (E. DETOMI, M. MORIGI, P. SHUMYATSKY, AL (2012)) Let G be a finite group. Then there exists a 3-generated subgroup H of G such that H has the same exponent of G.

33 EXPONENT THEOREM (E. DETOMI, M. MORIGI, P. SHUMYATSKY, AL (2012)) Let G be a finite group. Then there exists a 3-generated subgroup H of G such that H has the same exponent of G. A finite solvable group G contains a 2-generated subgroup H with the same exponent. We don t know whether this is true for arbitrary finite groups.

34 PROOF IN THE SOLVABLE CASE Assume that G is a finite solvable group without proper subgroups of the same exponent. There exist: 1 an elementary abelian p-group A 2 an irreducible subgroup H of Aut(A) 3 a positive integer k such that d(g) = d(a k H) > d(a k 1 H) 4 an epimorphism φ : G A k H. Let exp(g) = p a m with (p, m) = 1. Choose g G with g = p a : there exist a A k, h H with φ(g) = ah. Consider K = φ 1 ( a, H ). exp(g) = exp(k ) G = K A k H = a, H A k is a cyclic H-module d(g) = d(a k H) 2

35 PRIME DIVISORS OF INDICES THEOREM Let G be a finite group and let X, C be subgroups of G with X C. Then there exist a, b G s.t. π( G : C ) π( a, b, X : C a, b, X ). G C a, b, X C a, b, X X

36 When X = C we obtain: COROLLARY Let G be a finite group and let C be a subgroup of G. Then there exist a, b G such that π( G : C ) = π( a, b, C : C ). When X is the identity subgroup we obtain: COROLLARY Let G be a finite group and let C be a subgroup of G. Then there exist a, b G such that π( G : C ) π( a, b : C a, b ).

37 AN APPLICATION Let Ind G (x) be the index in G of the centralizer C G (x) of x in G. THEOREM (A.R CAMINA, P. SHUMYATSKY, C. SICA (2010)) If Ind a,b,x (x) is a prime-power for every a, b G, then Ind G (x) is a prime-power.

38 AN APPLICATION Let Ind G (x) be the index in G of the centralizer C G (x) of x in G. THEOREM (A.R CAMINA, P. SHUMYATSKY, C. SICA (2010)) If Ind a,b,x (x) is a prime-power for every a, b G, then Ind G (x) is a prime-power. This theorem can be restated: let C = C G (x) and X = x ; if there is more than one prime dividing G : C, then there exist a, b G such that a, b, X : C a, b, X is divisible by more than one prime. From our previous result we have a stronger result: there exist a, b G with π( G : C ) π( a, b, X : C a, b, X ). In other words:

39 AN APPLICATION Let Ind G (x) be the index in G of the centralizer C G (x) of x in G. THEOREM (A.R CAMINA, P. SHUMYATSKY, C. SICA (2010)) If Ind a,b,x (x) is a prime-power for every a, b G, then Ind G (x) is a prime-power. This theorem can be restated: let C = C G (x) and X = x ; if there is more than one prime dividing G : C, then there exist a, b G such that a, b, X : C a, b, X is divisible by more than one prime. From our previous result we have a stronger result: there exist a, b G with π( G : C ) π( a, b, X : C a, b, X ). In other words: THEOREM For each x G there exists a and b in G such that π(ind G (x)) π(ind x,a,b (x)).

40 Given a subset X of G, let d X (G) denote the minimum integer d such that there exist d elements g 1,..., g d G with the property that G = X, g 1,..., g d. THEOREM Let X be a subset of a finite group G and let N be a normal subgroup of G such that N is maximal with the property that d XN (G) = d X (G). Then there exist a monolithic primitive group L and an isomorphism ϕ : G/N L k such that ϕ(x) {(l,..., l) L k l L}.

Boundedly generated subgroups of profinite groups. Elisa Covato (University of Bristol) June, 2014

Boundedly generated subgroups of profinite groups. Elisa Covato (University of Bristol) June, 2014 Elisa Covato University of Bristol, UK Young Algebraists Conference 2014 EPFL, Lausanne June, 2014 Introduction Let G be a finite group and let i(g) be an invariant e.g. i(g) could be the set of prime