The probability of generating a soluble subgroup of a finite group

It is proved that for finite groups G, the probability thattwo randomly chosen elements of G generate a soluble subgrouptends to zero as the index of the largest soluble normal subgroupof G tends to infinity.     

The probability of generating a finite simple group

We show that two random elements of a finite simple groupG generateG with probability 1 as |G| . This settles a conjecture of Dixon.     

The probability of generating a finite simple group

We study the probability of generating a finite simple group, together with its generalisation P _G,socG (d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/socG. We prove that P _G,socG (2) ? 53/90, with equality if and only if G is A₆ or S₆, and establish a similar result for P _G,socG (3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel’nikov on profinite groups, follow easily from our results.     

The probability of generating a finite classical group

In this paper generalized Fischer spaces are defined and some results on their classification are obtained.     

The probability of generating a finite linear group

We study the asymptotic behavior of the probability of generating a finite completely reducible linear group G of degree n with [ n] elements. In particular we prove that if 3/2 and n is large enough then [ n] randomly chosen elements that generate G modulo O²(G) almost certainly generate G itself.Received: 13 February 2003     

The probability of generating a classical group

We show that a countable totally and discretely ordered set with first element inherently carries the structure of an ordered commutative euclidean monoid, provided its order type is of a certain kind. As an application we specify the order types of all discretely ordered sets which can be expanded to ordered commutative euclidean monoids.     

The probability of generating a permutation group

It is well known that a permutation group of degree can be generated by elements. In this paper we study the asymptotic behavior of the probability of generating a permutation group of degree n with elements. In particular we prove that if n is large enough and elements generate a permutation group G of degree n modulo G G ², then almost certainly these elements generate G itself. Received: 2 January 2002     

On solvability of groups with a finite nilpotent supercomplemented subgroup

We investigate groups in which every subgroup containing some fixed finite nilpotent subgroup has a complement.     

Autocommuting probability of a finite group

Let G be a finite group and Aut(G) the automorphism group of G. The autocommuting probability of G, denoted by Pr(G,Aut(G)), is the probability that a randomly chosen automorphism of G fixes a randomly chosen element of G. In this paper, we study Pr(G,Aut(G)) through a generalization. We obtain a computing formula, several bounds and characterizations of G through Pr(G,Aut(G)). We conclude the paper by showing that the generalized autocommuting probability of G remains unchanged under autoisoclinism.     

Comonolithic subnormal subgroups generating a finite group

Every finite group is generated by minimal sets of subnormal comonolithic subgroups all this sets having the same number of members and whose perfect subgroups being the same for every such a set. The proof of this result depends on some well known results about subnormal subgroups, mainly due to H. Wielandt.     

On products of two nilpotent subgroups of a finite group

LetG be a finite group with an abelian Sylow 2-subgroup. LetA be a nilpotent subgroup ofG of maximal order satisfying class (A)≦k, wherek is a fixed integer larger than 1. Suppose thatA normalizes a nilpotent subgroupB ofG of odd order. ThenAB is nilpotent. Consequently, ifF(G) is of odd order andA is a nilpotent subgroup ofG of maximal order, thenF(G)?A.