Let G be a finite group,
A and B two elements of G, which generate a subgroup L of order λ. We
call an expression of the form Aα1Bβ1Aα2⋯Bβ2 with αi,βi≧ 0 a word in
A and B and ∑i(αi+ βi) the weight of the word. For any g ∈ G define
fm(g) as the number of words of weight m which are equal to g. Our purpose
in this paper is to investigate the asymptotic dependence of fm(g) on m.
Subject to some simple side conditions, it turns out that the elements of L
all occur with relative equal frequency as m approaches infinity. We also
have an estimate of the smallest weight for which all elements of L can be
realized.
