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

HTML articles powered by AMS MathViewer

- by R. B. J. T. Allenby and C. Y. Tang PDF
- Trans. Amer. Math. Soc.
**203**(1975), 319-330 Request permission

## 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 $H$.## References

- R. B. J. T. Allenby,
*On the residual finiteness of permutational products of groups*, J. Austral. Math. Soc.**11**(1970), 504–506. MR**0276351**, DOI 10.1017/S1446788700007953 - R. B. J. T. Allenby and C. Y. Tang,
*On the Frattini subgroups of generalized free products*, Bull. Amer. Math. Soc.**80**(1974), 119–121. MR**327908**, DOI 10.1090/S0002-9904-1974-13381-1 - 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 permutational products of groups*, J. Austral. Math. Soc.**10**(1969), 111–135. MR**0245686**, DOI 10.1017/S1446788700006947 - 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 - 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 - B. H. Neumann,
*Permutational products of groups*, J. Austral. Math. Soc.**1**(1959/1960), 299–310. MR**0123597**, DOI 10.1017/S1446788700025970 - B. H. Neumann,
*On amalgams of periodic groups*, Proc. Roy. Soc. London Ser. A**255**(1960), 477–489. MR**113927**, DOI 10.1098/rspa.1960.0081 - C. Y. Tang,
*On the Frattini subgroups of generalized free products with cyclic amalgamations*, Canad. Math. Bull.**15**(1972), 569–573. MR**314989**, DOI 10.4153/CMB-1972-099-5 - C. Y. Tang,
*On the Frattini subgroups of certain generalized free products of groups*, Proc. Amer. Math. Soc.**37**(1973), 63–68. MR**310073**, DOI 10.1090/S0002-9939-1973-0310073-6 - Alice Whittemore,
*On the Frattini subgroup*, Trans. Amer. Math. Soc.**141**(1969), 323–333. MR**245687**, DOI 10.1090/S0002-9947-1969-0245687-9

## Additional Information

- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**203**(1975), 319-330 - MSC: Primary 20E30
- DOI: https://doi.org/10.1090/S0002-9947-1975-0357616-0
- MathSciNet review: 0357616