Let G be a finite group; there exists a uniquely determined
Dirichlet polynomial P(G,s) such that if t is a positive integer, then
P(G,t) gives the probability of generating G with t randomly
chosen elements. We show that it may be recognized from the knowledge of P(G,s) whether G/Frat(G) iis a simple group
This paper provides a comprehensive investigation of the cellular approximation functor cell A G, in the category of groups, approximating a group G by a group A. We also study related notions such as A-injection, A-generation and A-constructibility of a group G and we find several interesting connections with the Schur multiplier H 2 (G, Z). Our constructions are direct and are given in a slow and detailed manner.
In the factorial ring of Dirichlet polynomials we explore the connections between how the Dirichlet polynomial P G (s) associated with a finite group G factorizes and the structure of G. If P G (s) is irreducible, then G/Frat G is simple. We investigate whether the converse is true, studying the factorization in the case of some simple groups. For any prime p ≥ 5 we show that if P G (s) = P Alt(p) (s), then G/Frat G ∼ = Alt(p) and P Alt(p) (s) is irreducible. Moreover, if P G (s) = P PSL(2,p) (s), then G/Frat G is simple, but P PSL(2,p) (s) is reducible whenever p = 2 t − 1 and t = 3 mod 4.
Abstract. Let G be a finite group; there exists a uniquely determined Dirichlet polynomial P G (s) such that if t ∈ ,ގ then P G (t) gives the probability of generating G with t randomly chosen elements. We show that if2000 Mathematics Subject Classification. 20P05, 20D06.
We find a formula for computing the minimum number of generators for the augmentation ideal in the profinite setting; this is a generalisation of a result obtained in the finite case by Cossey, Gruenberg and Kovács.We then consider probabilistic questions connected with the generation of the augmentation ideal and we define the class of APFG groups as those profinite groups for which the probability of generating the augmentation ideal with t random elements is non-zero for some t ∈ N. We give a characterisation of these groups which allows us easily to compare APFG groups with PFG groups and we show that PFG groups are APFG but we give examples showing that this is a strict inclusion.
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