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On the Isomorphism of Integral Group Rings. I

Published online by Cambridge University Press:  20 November 2018

Sudarshan K. Sehgal
Sudarshan K. Sehgal*
Affiliation:
Ohio State University, Columbus, Ohio
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In this note we study the question of automorphisms of the integral group ring Z(G) of a finite group G. We prove that if G is nilpotent of class two, any automorphism of Z(G) is composed of an automorphism of G and an inner automorphism by a suitable unit of Q(G), the group algebra of G with rational coefficients. In § 3, we prove that if two finitely generated abelian groups have isomorphic integral group rings, then the groups are isomorphic. This is an extension of the classical result of Higman (2) for the case of finite abelian groups. In the last section we give a new proof of the fact that an isomorphism of integral group rings of finite groups preserves the lattice of normal subgroups. Other proofs are given in (1;4).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 410 - 413
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Cohn, J. A. and Livingstone, D., On the structure of group algebras. I, Can. J. Math. 17 (1965), 583–593 Google Scholar
2. Higman, G., The units of group rings, Proc. London Math. Soc. 46 (1940), 231–248.Google Scholar
3. Jackson, D. A., Ph.D. Thesis, Oxford University, Oxford, 1967.Google Scholar
4. Passman, D. S., Isomorphic groups and group rings, Pacific J. Math. 15 (1965), 561–583.Google Scholar
5. van der Waerden, B. L., Modern algebra (Ungar, New York, 1950).Google Scholar
6. Whitcomb, A., Ph.D. Thesis, University of Chicago, Chicago, Illinois, 1967.Google Scholar