Federico Menegazzo

Asymptotic Results for Transitive Permutation Groups

In this paper we give answers to some open questions concerning generation and enumeration of finite transitive permutation groups. In [1], Bryant, Kovács and Robinson proved that there is a number c′ such that each soluble transitive permutation group of degree n > 2 can be generated by [c′n/ √ log n ] elements, and later A. Lucchini [5] extended this result (with a different constant c′) to finite permutation groups containing a soluble transitive subgroup. We are now able to prove this theorem in full generality, and this solves the question of bounding the number of generators of a finite transitive permutation group in terms of its degree. The result obtained is the following.

Groups with trivial virtual automorphism group

We answer a question of A. Lubotzky and A. Mann by constructing examples of infinite groupsG such that every isomorphismα:H →K between subgroupsH andK having finite index inG coincides with the identity on some subgroup of finite index. The structure of such a group is very restricted;G must be virtually a 2-group with finite central derived subgroup andG/G′ elementary abelian.

ON THE PROBABILITY OF GENERATING PROSOLUBLE GROUPS

LetF be the free prosoluble group of rankd. We determine the minimum integerk such that the probability of generatingF withk elements is positive.

Computing a Set of Generators of Minimal Cardinality in a Solvable Group

In this paper two algorithms are presented which compute a set of generators of minimal cardinality for a finite soluble group given by a polycyclic presentation. The first can be used when a chief series is available. The second algorithm is less simple, but nevertheless efficient, and can be used when it is difficult or too expensive to compute a chief series. The problem of determining the minimal number d(G) of generators when G is a solvable group has been discussed and solved by Gaschutz, and the ideas for these algorithms are essentially suggested by the work of Gaschutz. In the Appendices CAYLEY V.3.7.2 procedures are listed.

Complements of the socle in monolithic groups

We show that if H is a finite group with a unique minimal normal subgroup N , which is not abelian, then the number of conjugacy classes of complements of N in H is strictly smaller than jN j.

Generating Permutation Groups

Abstract In this paper an algorithm is produced, which, given a permutation group G of degree n > 3, outputs a generating set for G with at most n/2 elements.

On the composition length of finite primitive linear groups

LetG be a finite primitive linear group over a fieldK, whereK is a finite field or a number field. We bound the composition length ofG in terms of the dimension of the underlying vector space and of the degree ofK over its prime subfield. As a byproduct, we prove a result of number theory which bounds the number of prime factors (counting multiplicities) ofqn−1, whereq, n>1 are integers, improving a result of Turull and Zame [6].