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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the Frattini subgroup of a residually finite generalized free product


Authors: R. B. J. T. Allenby and C. Y. Tang
Journal: Proc. Amer. Math. Soc. 47 (1975), 300-304
MSC: Primary 20E30
MathSciNet review: 0390066
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Abstract: Let $ G = {(A \ast B)_H}$ be the generalized free product of the groups $ A,B$ amalgamating the subgroup $ H$, and let $ \Phi (G)$ denote its Frattini subgroup. In support of the conjecture that $ \Phi (G) \subseteq H$ whenever $ G$ is resiually finite and $ H$ satisfies a nontrivial identical relation, we show, amongst several other things, that the above inequality is indeed valid if in addition at least one of the following holds: (i) $ A,B$, each satisfies a nontrivial identical relation; (ii) $ G$ is finitely generated; (iii) $ H$ is nilpotent. In particular (i) completes earlier investigations of the second author. The methods of proof are, however, different.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0390066-5
Keywords: Frattini subgroup, residually finite, generalized free product, identical relation
Article copyright: © Copyright 1975 American Mathematical Society