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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the Frattini subgroups of generalized free products and the embedding of amalgams

Authors: R. B. J. T. Allenby and C. Y. Tang
Journal: Trans. Amer. Math. Soc. 203 (1975), 319-330
MSC: Primary 20E30
MathSciNet review: 0357616
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Abstract: In this paper we shall prove a basic relation between the Frattini subgroup of the generalized free product of an amalgam $ \mathfrak{A} = (A,B;H)$ and the embedding of $ \mathfrak{A}$ into nonisomorphic groups, namely, if $ \mathfrak{A}$ can be embedded into two non-isomorphic groups $ {G_1} = \langle A,B\rangle $ and $ {G_2} = \langle A,B\rangle $ then the Frattini subgroup of $ G = {(A \ast B)_H}$ is contained in $ H$. We apply this result to various cases. In particular, we show that if $ A,B$ are locally solvable and $ H$ is infinite cyclic then $ \Phi (G)$ is contained in $ H$.

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Keywords: Frattini subgroup, free group, free product, generalized free product, permutational product, wreath product, nilpotent group, solvable group, amalgamated subgroup, amalgam, identical relation, $ G$-normal
Article copyright: © Copyright 1975 American Mathematical Society