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Notes on Frattini Subgroups of Generalized Free Products with Cyclic Amalgamation

Published online by Cambridge University Press:  20 November 2018

R. B. J. T. Allenby ,
C. Y. Tang  and
S. Y. Tang
R. B. J. T. Allenby
Affiliation:
School of Mathematics, University of Leeds, Leeds, England
C. Y. Tang
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1
S. Y. Tang
Affiliation:
Department of Mathematics, San Francisco State University, California, U.S.A.
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The problem of the exact location of the Frattini subgroup 4>(G) of a generalized free product G = (A*B)H was first raised by Higman and Neumann [5]. Solutions to special cases of the problem can be found in [1], [2], [8], [9] and [10]. The purpose of this note is to extend the results of [2], [8], and to simplify the proof of Whittemore's theorem [10]. We also apply our result to give simple proofs of certain classes of knot groups that have trivial Frattini subgroups. The proof that every knot group has trivial Frattini subgroup hard and long (footnote 2, p. 56).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 23 , Issue 1 , 01 March 1980 , pp. 51 - 59
Copyright
Copyright © Canadian Mathematical Society 1980

References

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