Covering Theorems in the Alternating Group

Abstract

It was shown by Ore [1951] that each element of A n, the alternating group on n letters, is a commutator, aba-lb-1 for some a and b in S n. Xu (1965] went further and showed that, given an element a in A n then, given any gEAn there is some b in A n such that g is conjugate to aba-lb-1. using the result of Xu it can be shown that there exist conjugacy classes C and C' in An such that CC' = A n. In this paper we will look at the case where C = C' and determine whether some simple classes have the property CC = A n. Section 2 looks at classes whose elements have period two, utilizing results from Brenner, Randall and Riddell [1974]. Section three looks at classes whose elements have period three and contains results of Brenner and Riddell [1976]. Section four looks at coverings by classes whose elements all have period n (n odd) or period n-1 (n even) and section five looks at the closely related problem when, for n = 16k, the elements of the conjugacy classes all have period n/4 using results from Brenner, Cranwell and Riddell [1973] and [1974]. This paper then will summarize and complete details for several different types of proofs.