Abstract

To measure the commutativity of two elements in a group, we define the commutator by [x, y] = x -1 y -1 xy for all x, y G. Two elements x, y G commute if and only if [x, y] = e, where e is the identity element of G.We determine the number of pairs (x, y) in a finite group such that [x, y] = e.Using the free group of rank n, denoted F n , we define for every g G the set of n-tuples h G n such that w(h) = g, denoted by N G, w (g).Many open problems remain about the structure of N G, w (g) and its implications for the underlying group.One such problem is Amit's Conjecture, which states that for every w F n , the value of |N G, w (e)| is at least |G| n-1 for every finite nilpotent group G.This paper proves Amit's Conjecture for all words in two variables.Furthermore, our main theorem generalizes the result to all finite groups.As an application, we provide a bound on the number of edges in the non-braid graph of any finite nonabelian group.

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