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Semi-Homomorphisms of Groups

Published online by Cambridge University Press:  20 November 2018

I. N. Herstein
I. N. Herstein*
Affiliation:
University of Chicago, Chicago, Ill.
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A mapping ϕ from one group, G, into another, H, is said to be a semi-homomorphism of G if ϕ(aba) = ϕ(a) ϕ(b) ϕa) for all a, b ∊G. Clearly any homomorphism or anti-homomorphism is a semi-homomorphism; the converse, however, need not be true in general. It is perfectly clear what one intends by a semi-isomorphism or semi-automorphism.

Our purpose here is to show that for a rather general situation a semi-homomorphism turns out to be a homomorphism or an anti-homomorphism. In (2) we proved that any semi-automorphism of a simple group which contains an element of order 4 must automatically be either an automorphism or an anti-automorphism.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 384 - 388
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

In the first version of this paper the results were proved for the case of semi-automorphisms of simple groups. The results were generalized to the present situation while the author was a guest at the Mathematics Research Institute of the E.T.H. in Zürich.

This work was supported by a grant from the Army Research Office, ARO(D), at the University of Chicago.

References

1. Feit, Walter and Thompson, John, Solvability of groups of odd order, Pacific J. Math., 13 1963), 775–1029.Google Scholar
2. Herstein, I. N. and Ruchte, M. F., Semi-automorphisms of groups, Proc. Amer. Math. Soc, 9 (1958), 145–150.Google Scholar