On the probability of generating a monolithic group

Eloisa Detomi, Andrea Lucchini, Colva Mary Roney-Dougal

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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.

Original languageEnglish
Pages (from-to)413-433
JournalJournal of Algebra
Volume407
Early online date18 Apr 2014
DOIs
Publication statusPublished - 1 Jun 2014

Keywords

  • Finite group theory
  • Random generation of finite groups
  • Finite simple groups
  • Crown-based power

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