On the Frattini subgroups of certain generalized free products of groups

Tang, C. Y.

doi:10.1090/S0002-9939-1973-0310073-6

On the Frattini subgroups of certain generalized free products of groups
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by C. Y. Tang

Proc. Amer. Math. Soc. 37 (1973), 63-68

DOI: https://doi.org/10.1090/S0002-9939-1973-0310073-6

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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 ${H^G}$.

References

Gilbert Baumslag, On the residual finiteness of generalised free products of nilpotent groups, Trans. Amer. Math. Soc. 106 (1963), 193–209. MR 144949, DOI 10.1090/S0002-9947-1963-0144949-8
D. Ž. Djoković and C. Y. Tang, On the Frattini subgroup of the generalized free product with amalgamation, Proc. Amer. Math. Soc. 32 (1972), 21–23. MR 289656, DOI 10.1090/S0002-9939-1972-0289656-7
Joan Landman Dyer, On the residual finiteness of generalized free products, Trans. Amer. Math. Soc. 133 (1968), 131–143. MR 237649, DOI 10.1090/S0002-9947-1968-0237649-1
R. J. Gregorac, On residually finite generalized free products, Proc. Amer. Math. Soc. 24 (1970), 553–555. MR 260878, DOI 10.1090/S0002-9939-1970-0260878-2
Graham Higman and B. H. Neumann, On two questions of Itô, J. London Math. Soc. 29 (1954), 84–88. MR 57881, DOI 10.1112/jlms/s1-29.1.84
Abraham Karrass and Donald Solitar, On finitely generated subgroups of a free group, Proc. Amer. Math. Soc. 22 (1969), 209–213. MR 245655, DOI 10.1090/S0002-9939-1969-0245655-2
B. H. Neumann, An essay on free products of groups with amalgamations, Philos. Trans. Roy. Soc. London Ser. A 246 (1954), 503–554. MR 62741, DOI 10.1098/rsta.1954.0007
Alice Whittemore, On the Frattini subgroup, Trans. Amer. Math. Soc. 141 (1969), 323–333. MR 245687, DOI 10.1090/S0002-9947-1969-0245687-9