Trivial action on the tensor product of finite groups
Glasgow mathematical journal, Tome 30 (1988) no. 3, pp. 271-274
Voir la notice de l'article provenant de la source Cambridge University Press
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 G ⊗ H by (G ⊗ H)K. In section 1 of this note we shall obtain necessary and sufficient conditions for (G ⊗ H)K = G ⊗ H. 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) × (G ⊗ H); 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) ≅ G ∧ G to find necessary and sufficient conditions for M(G)K = M(G).
Higgs, R. J. Trivial action on the tensor product of finite groups. Glasgow mathematical journal, Tome 30 (1988) no. 3, pp. 271-274. doi: 10.1017/S0017089500007357
@article{10_1017_S0017089500007357,
author = {Higgs, R. J.},
title = {Trivial action on the tensor product of finite groups},
journal = {Glasgow mathematical journal},
pages = {271--274},
year = {1988},
volume = {30},
number = {3},
doi = {10.1017/S0017089500007357},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500007357/}
}
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