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The number of solutions of xp = 1 in a finite group

Published online by Cambridge University Press:  24 October 2008

Thomas J. Laffey
Thomas J. Laffey
Affiliation:
University College, Dublin
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Extract

Let G be a finite group, p a prime divisor of |G| and suppose that G is not a p-group. In this note, we show that the number of elements x ∈ G such that xp = 1 is at most (p|G|)/(p + 1). This answers a question posed by D. MacHale. When G is a Frobenius group of order p(p + 1), p a Mersenne prime, the above bound is attained.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 80 , Issue 2 , September 1976 , pp. 229 - 231
Copyright
Copyright © Cambridge Philosophical Society 1976

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References

REFERENCES

(1)Gorenstein, D.Finite groups (New York, Harper and Row, 1968).Google Scholar
(2)Liebeck, H. and MacHale, D.Groups with automorphisms inverting most elements. Math. Z. 124 (1972), 51–63.CrossRefGoogle Scholar