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Proceedings of the American Mathematical Society

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

 

 

On the Frattini subgroups of certain generalized free products of groups


Author: C. Y. Tang
Journal: Proc. Amer. Math. Soc. 37 (1973), 63-68
MSC: Primary 20F25
DOI: https://doi.org/10.1090/S0002-9939-1973-0310073-6
MathSciNet review: 0310073
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Abstract: Let $ G = (\prod\nolimits_{i \in I}^\ast {{A_i}{)_H}} $ be the generalized free product of the groups $ {A_i}$ amalgamating the subgroup H. We show that if G is residually finite and the groups $ {A_i}$ have compatible H-filters then the Frattini subgroup $ \Phi (G)$ is contained in the maximal G-normal subgroup in H. If the groups $ {A_i}$ are free and H is finitely generated of infinite index in one $ {A_i}$ then $ \Phi (G) = 1$. We also show that if H is simple then $ \Phi (G) = 1$ or $ {H^G}$.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0310073-6
Keywords: Frattini subgroup, free group, simple group, generalized free product, amalgamated subgroup, residually finite, H-filter, G-normal, identical relation
Article copyright: © Copyright 1973 American Mathematical Society