Soluble Groups with Finite Wielandt length

A Problem of Wielandt on Finite Permutation Groups

Problem 6.6 in the Kourovka Notebook [9], posed by H. Wielandt, reads as follows. ' Let P, Q be permutation representations of a finite group G with the same character. Suppose P(G) is a primitive permutation group. Is Q{G) necessarily primitive? Equivalently: Let A, B be subgroups of a finite group G such that for each class C of conjugate elements of G their intersection with C has the same c...

Constructing Normalisers in Finite Soluble Groups

This paper describes algorithms for constructing a Hall n-subgroup H of a finite soluble group G and the normaliser No(H). If G has composition length n, then H and No(H ) can be constructed using O(n ~ log IGI) and O(n ~ log IGI) group multiplications, respectively. These algorithms may be used to construct other important subgroups such as Carter subgroups, system normalisers and relative sys...

Computing automorphisms of finite soluble groups

There is a large collection of e ective algorithms for computing information about nite soluble groups. The success in computation with these groups is primarily due to a computationally convenient representation of them by means of (special forms of) power conjugate presentations. A notable omission from this collection of algorithms is an e ective algorithm for computing the automorphism grou...

On Minimal Length Factorizations of Finite Groups

Length of polynomials over finite groups

We study the length of polynomials over nite simple non-Abelian groups needed to realize Boolean functions. We apply the results for bounding the length of 5-permutation branching programs recognizing a Boolean set. Moreover, for Boolean and general functions on these groups, we present upper bounds on the length of shortest polynomials computing an arbitrary nary Boolean or general function, o...