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

Author:

Allenby R. B. J. T.,Tang C. Y.

Abstract

In this paper we shall prove a basic relation between the Frattini subgroup of the generalized free product of an amalgam A = ( A , B ; H ) \mathfrak {A} = (A,B;H) and the embedding of A \mathfrak {A} into nonisomorphic groups, namely, if A \mathfrak {A} can be embedded into two non-isomorphic groups G 1 = A , B {G_1} = \langle A,B\rangle and G 2 = A , B {G_2} = \langle A,B\rangle then the Frattini subgroup of G = ( A B ) H G = {(A \ast B)_H} is contained in H H . We apply this result to various cases. In particular, we show that if A , B A,B are locally solvable and H H is infinite cyclic then Φ ( G ) \Phi (G) is contained in H H .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference14 articles.

1. On the residual finiteness of permutational products of groups;Allenby, R. B. J. T.;J. Austral. Math. Soc.,1970

2. On the Frattini subgroups of generalized free products;Allenby, R. B. J. T.;Bull. Amer. Math. Soc.,1974

3. On the residual finiteness of generalised free products of nilpotent groups;Baumslag, Gilbert;Trans. Amer. Math. Soc.,1963

4. On the Frattini subgroup of the generalized free product with amalgamation;Djoković, D. Ž.;Proc. Amer. Math. Soc.,1972

5. On the residual finiteness of generalized free products;Dyer, Joan Landman;Trans. Amer. Math. Soc.,1968

Cited by 3 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Infinite groups;Journal of Soviet Mathematics;1982

2. The solutions to two problems on permutational products;Journal of the Australian Mathematical Society;1981-12

3. On the frattini subgroups of generalized free products;Journal of Algebra;1978-06

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