Nielsen transformations

Abstract

This chapter discusses application of Nielsen transformations. It presents an assumption involving a group G with n generators, ∑ a set of ordered sets of n generators of G, and the group A of permutations of ∑ generated by all the above. The chapter presents the problem: If G has n − 1 generators, then in every transitivity class of ∑ under A, is there a set of generators one of which is the unit element? The answer to this question is yes if G is finite and soluble. If G is a finite soluble group with n −1 generators, then A is transitive on ∑. To find a counterexample for the nonsoluble case, a computer might be employed. If G is the alternating group on five symbols and ∑ is the set of sets of three elements that generate G, then ∑ has 120 × 1668 elements. These are partitioned into 1,668 transitivity classes under the action of automorphisms of G, and A can be regarded as acting on these classes rather than on the elements of ∑.