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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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On the Frattini subgroups of generalized free products and the embedding of amalgams
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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 $H$.
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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