1,721,004 research outputs found
Base sizes for simple groups and a conjecture of Cameron
Full text linkLet G be a permutation group on a finite set ?. A base for G is a subset B C_ ? whose pointwise stabilizer in G is trivial; we write b(G) for the smallest size of a base for G. In this paper we prove that b(G) ? if G is an almost simple group of exceptional Lie type and is a primitive faithful G-set. An important consequence
of this result, when combined with other recent work, is that b(G) ? 7 for any almost simple group G in a non-standard action, proving a conjecture of Cameron. The proof is probabilistic and uses bounds on fixed point ratios
Characterization of p-adic analytic groups in terms of wreath products
No full textAbstractWe show that a finitely generated pro-p group is p-adic analytic (i.e., can be given the structure of a Lie group over Qp) if and only if it does not involve arbitrarily large wreath products of the form Cp wr Cpn. This result, whose proof applies Zelmanov's recent solution to the restricted Burnside problem, is in fact equivalent to Zelmanov's Theorem
Mixing and generation in simple groups
No full textAbstractLet G be a finite simple group. We show that a random walk on G with respect to the conjugacy class xG of a random element x∈G has mixing time 2. In particular it follows that (xG)2 covers almost all of G, which could be regarded as a probabilistic version of a longstanding conjecture of Thompson. We also show that if w is a non-trivial word, then almost every pair of values of w in G generates G
Meta-abelian unit groups of group algebras are usually abelian
No full textAbstractGiven a field F of positive characteristic p>2 and a finite group G we give necessary and sufficient conditions for the unit group of the group algebra FG to be meta-abelian. In particular, if p⩾5, this happens only when G is abelian
Dimension subgroups, nilpotency indices, and the number of generators of ideals in p-group algebras
No full textAbstractVarious problems in modular p-group algebras are solved through extensive study of dimension subgroups. As an application it is shown that, if G is an infinite res-P group (p > 2) and K is a field of characteristic p, then the least upper bound on the numbers of generators of ideals in KG equals the minimal index of a cyclic Subgroup of G
The derived length of Lie soluble group rings I
No full textAbstractLet KG be a Lie soluble group ring of positive characteristic. Some results relating the derived length of KG to the structure of G are described. In particular, it is shown that, if the derived length of KG is at most n and char(K) > 2n, then G is abelian
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