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On P. Hall's Generalisation of a Theorem of Frobenius

Published online by Cambridge University Press:  18 May 2009

S. K. Sehgal
S. K. Sehgal
Affiliation:
University of Notre Dame, Notre Dame, Indiana, U.S.A.
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1. It is a well known theorem due to Frobenius that the number of solutions of the equation

Xn = 1

in a finite group G, is a multiple of the greatest common divisor (n, g) of n and the order g of G. Frobenius himself proved later that the number of solutions of the equation

Xn = a

where a is a fixed element of G, is a multiple of (n, ga), ga being the order of the centralizer Z(a) of ain G.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 5 , Issue 3 , January 1962 , pp. 97 - 100
Copyright
Copyright © Glasgow Mathematical Journal Trust 1962

References

1.Hall, P., On a theorem of Frobenius, Proc. London Math. Soc. 40 (1936), 468–501.CrossRefGoogle Scholar
2.Prokofyev, A. N., On the fundamental theorem of Frobenius, Doklady Akad. Nauk SSSR (N.S.) 65 (1949), 801–804.Google Scholar