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Trivial action on the tensor product of finite groups

  • R. J. Higgs (a1)

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Let G, H and K be finite groups such that K acts on both G and H. The action of K on G and H induces an action of K on their tensor product G ⊗ H, and we shall denote the K-stable subgroup of GH by (GH)K. In section 1 of this note we shall obtain necessary and sufficient conditions for (GH)K = GH. The importance of this result is that the direct product of G and H has Schur multiplier M(G × H) isomorphic to M(G) × M(H) × (GH); moreover K: acts on M(G × H), and M(G × H)K is one of the terms contained in a fundamental exact sequence concerning the Schur multiplier of the semidirect product of K and G × H (see [3, (2.2.10) and (2.2.5)] for details). Indeed in section 2 we shall assume that G is abelian and use the fact that M(G) ≅ GG to find necessary and sufficient conditions for M(G)K = M(G).

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References

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1.Gorenstein, D., Finite groups, (Chelsea, New York, 1980).
2.Huppert, B., Endliche Gruppen I, (Springer, 1967).
3.Karpilovsky, G., The Schur multiplier, (Oxford University Press, 1987).
4.Maclane, S., Homology, (Springer, 1963).

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