The lower bound of diameter of Alternating groups

Abstract

In this paper we consider a specific case of the diameter search problem for finite groups, thecase where the system of generators is fixed. This problem is well-known and can be formulated in the following way: find the diameter of a group over its system of generators. The diameter of the corresponding Cayley graph is the diameter of a group over its specific system of generators. The main object of the research is the alternating group with the system of generators consisting of cycles having length three and the form (1,2,k). This system of generators is a classical irreducible system of generators of the alternating group. It is introduced the property of even permutations to be balanced. We consider the set of balanced permutations and permutations close enough to balanced and find minimum decompositions of them over defined system of generators. The main result of the paper is the lower bound of the diameter of Alternating group over con-sidered system of generators. The estimation is achieved using minimal decompositions of balanced permutations.