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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, <IMG WIDTH="22" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$G$">-normal
Article copyright: © Copyright 1975 American Mathematical Society