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

Allenby, R. B. J. T.; Tang, C. Y.

doi:10.1090/S0002-9947-1975-0357616-0

On the Frattini subgroups of generalized free products and the embedding of amalgams
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by R. B. J. T. Allenby and C. Y. Tang

Trans. Amer. Math. Soc. 203 (1975), 319-330

DOI: https://doi.org/10.1090/S0002-9947-1975-0357616-0

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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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