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

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1 Elisa Covato University of Bristol, UK Young Algebraists Conference 2014 EPFL, Lausanne June, 2014

2 Introduction Let G be a finite group and let i(g) be an invariant e.g. i(g) could be the set of prime divisors of G, or the exponent of G Question Is there a subgroup H of G such that: H is generated by few elements; and i(h) = i(g)?

3 The prime graph Let π(g) be the set of prime divisors of G. The prime graph of G, denoted Γ(G), has vertex set π(g), and p q if and only if G contains an element of order pq.

4 The prime graph: An example Let G = A 10 be the alternating group of degree 10, so G = 10! 2 =

5 The prime graph Let π(g) be the set of prime divisors of G. The prime graph of G, denoted Γ(G), has vertex set π(g), and p q if and only if G contains an element of order pq. Theorem (Lucchini, Morigi & Shumyatsky, 2012) Let G be a finite group. Then G contains a 3-generated subgroup H such that Γ(H) = Γ(G).

6 The exponent Let exp(g) = lcm{ g : g G} denote the exponent of G. Theorem (Detomi & Lucchini, 2013) Let G be a finite group. Then G contains a 3-generated subgroup H such that exp(g) = exp(h). If G is solvable, then there exists a 2-generated subgroup H with the same exponent of G.

7 Profinite groups A profinite group is a compact and totally disconnected topological group (or, equivalently, it is an inverse limit of finite groups). Examples Finite groups The group of p-adic integers: Z p = lim Z/Z p n SL n (Z p ) Every Galois group is a profinite group A supernatural number is an infinite product p n(p) over all primes p, where 0 n(p). If G is a profinite group, then G := lcm{ G : N : N o G} is a supernatural number.

8 Main problem Problem Let G be a profinite group and let i(g) be an invariant. Is there a closed subgroup H of G such that H is generated by few elements; and i(h) = i(g)? A profinite group G is d-generated if and only if there exist x i G such that G = x 1,..., x d.

9 Results: The prime graph Theorem (E. Covato, 2014) Let G be a profinite group. Then G contains a 3-generated (closed) subgroup H such that Γ(H) = Γ(G). The main ingredient is the following result for finite groups: Proposition Let M be a finite group and let 1 = K 0... K n = M be a normal series for M. Then there exists a 3-generated subgroup L of M such that Γ(LK i /K i ) = Γ(M/K i ) for any i.

10 The exponent Problem A Is there a constant d such that every profinite group G contains a closed d-generated subgroup H such that exp(h) = exp(g)? This is related to the following well-known open problem: Problem B (1980) Does every torsion profinite group have finite exponent? A partial answer was provided by Zel manov: Theorem (Zel manov, 1992) Every finitely generated pro-p torsion group is finite. This implies that a positive answer to Problem A will solve Problem B.

11 Results: The exponent A group is supersolvable if it has a normal series with cyclic factors. A prosupersolvable group is an inverse limit of finite supersolvable groups. Zel manov s Theorem allows us to give a best-possible solution to Problem A for finitely generated prosupersolvable groups: Theorem (E. Covato, 2014) Let G be a finitely generated prosupersolvable group. Then G contains a closed 2-generated subgroup H such that exp(h) = exp(g).

12 Strategy (prime graphs) There are four main steps: (1) Take a chain of countably many open normal subgroups of G: G = M 0 > M 1 > M 2 > > {1} (2) Choosing d N appropriately, for j N 0 define Ω j = {(x 1,..., x d ) G d : Γ( x 1 M j,..., x d M j ) = Γ(G/M j )} (3) Establish the existence of (x 1,..., x d ) j 0 Ω j (4) H = x 1,..., x d = Γ(H) = Γ(G) To prove (3), we use the compactness of G, together with the theory of crown-based powers for finite groups.

13 Open Problems Problem A. Is there a constant d such that every profinite group G contains a closed d-generated subgroup H such that exp(h) = exp(g)? DL Theorem. Let G be a finite group. Then G contains a 3-generated subgroup H such that exp(g) = exp(h). Solve Problem A! (Or at least solve it for additional families of profinite groups.) Does the DL Theorem hold with d = 2? No counterexamples are known, and it is true for solvable groups. Investigate similar bounded generation problems for other natural group invariants, such as the spectrum of a group.

14 Grazie!