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 be the generalized free product of the groups amalgamating the subgroup , and let denote its Frattini subgroup. In support of the conjecture that whenever is resiually finite and 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) , each satisfies a nontrivial identical relation; (ii) is finitely generated; (iii) 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