Abstract
A group L is primitive monolithic if L has a unique minimal normal subgroup, N, and trivial Frattini subgroup. By PL,N(k) we denote the conditional probability that k randomly chosen elements of L generate L, given that they project onto generators for L/N. In this article we show that PL,N(k) is controlled by PY,S(2), where N≅Sr and Y is a 2-generated almost simple group with socle S that is contained in the normalizer in L of the first direct factor of N. Information about PL,N(k) for L primitive monolithic yields various types of information about the generation of arbitrary finite and profinite groups.
Highlights
A group L is primitive monolithic if L has a unique minimal normal subgroup N, and trivial Frattini subgroup
By PL,N (k) we denote the conditional probability that k randomly chosen elements of L generate L, given that they project onto generators for L/N, and by d(L) we denote the cardinality of the smallest generating set for L
The results presented there depend on a detailed analysis of the behavior of PL,N (d(L)) when L is almost simple [14]. These bounds were strong enough to solve a series of open problems on the generation of finite groups, an interesting question arises from [14] and [6]: is it true that if k d(L) PL,N (k) is bounded below by the conditional probability PY,S(2)? Here Y is a 2-generated group such that S P Y NL(S1)/CL(S1) Aut S, with N ∼= S1 ×· · ·×Sr ∼= Sr
Summary
Eloisa Detomi a, Andrea Lucchini a, Colva M. Roney-Dougal b,∗,1 a Università degli Studi di Padova, Dipartimento di Matematica, Via Trieste 63, 35121 Padova, Italy b University of St Andrews, Mathematical Institute, St Andrews, Fife KY16 9SS, Scotland, United Kingdom article info. Article history: Received 20 September 2013 Available online 18 April 2014 Communicated by E.I. Khukhro. A group L is primitive monolithic if L has a unique minimal normal subgroup, N , and trivial Frattini subgroup. In this article we show that PL,N (k) is controlled by PY,S (2), where N ∼= Sr and Y is a 2-generated almost simple group with socle S that is contained in the normalizer in L of the first direct factor of N. Information about PL,N (k) for L primitive monolithic yields various types of information about the generation of arbitrary finite and profinite groups
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