Vol. 17, No. 2, 1966

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Asymptotic properties of groups generation

Oscar S. Rothaus

Vol. 17 (1966), No. 2, 319–322

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 Aα1Bβ1Aα2Bβ2 with αii 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.

Mathematical Subject Classification
Primary: 20.10
Received: 17 December 1964
Published: 1 May 1966
Oscar S. Rothaus