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Some computations in the modular representation ring of a finite group

Published online by Cambridge University Press:  24 October 2008

John Santa Pietro
John Santa Pietro
Affiliation:
Stevens Institute of Technology, Hoboken, N.J. 07030, U.S.A.
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Extract

Let p be an odd prime and G = HB be a semi-direct product where H is a cyclic, p-Sylow subgroup and B is finite Abelian. If K is a field of characteristic p the isomorphism classes of KG-modules relative to direct sum and tensor product generate a ring a(G) called the representation ring of G over K. If K is algebraically closed it is shown in (4) that there is a ring isomorphism a(G) ≃ a(HB2)⊗a(B1) where B1 is the kernel of the action of B on H and B2 = B/B1.> 2, Aut (H) is cyclic thus HB2 is metacyclic. The study of the multiplicative structure of a(G) is thus reduced to that of the known rings a(B1) and a(HB2) (see (3)).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 69 , Issue 1 , January 1971 , pp. 163 - 166
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

REFERENCES

(1)Feit, W.Groups with a cyclic Sylow subgroup. Nagoya Math. J. 27 (1966), 571–584.CrossRefGoogle Scholar
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(3)O'reilly, M. F.On the modular representation algebra of metacyclic groups. J. London Math. Soc. 39 (1964), 267–276.CrossRefGoogle Scholar
(4)Santa Pietro, J. The modular representation ring of a semi-direct product of finite groups. Ph.D. Thesis, Stevens Institute of Technology, 1969.Google Scholar