Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T15:59:48.573Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Allenby, R. B. J. T. and Tang, C. Y., On the Frattini subgroup of a residually finite generalized free product, Proc. Amer. Math. Soc., 47 (1975), 300-304.Google Scholar
2. Allenby, R. B. J. T. and Tang, C. Y., On the Frattini subgroups of generalized free products and the embedding of amalgams, Trans. Amer. Math. Soc., 203 (1975), 319-330.Google Scholar
3. James, Boler and Benny, Evans, The free product of residually finite groups amalgamated along retracts is residually finite, Proc. Amer. Math. Soc., 37 (1973) 50-52.Google Scholar
4. Gregorac, R. J., On residually finite generalized free products, Proc. Amer. Math. Soc., 24 (1970) 553-555.Google Scholar
5. Higman, G. and Neumann, B. H., On two questions of Ito, J. London Math. Soc., 29 (1954) 84-88.Google Scholar
6. Murasugi, K., Lecture notes on knot theory, Univ. of Toronto, 1971.Google Scholar
7. Neuwirth, L. P., Knot groups, Princeton University Press, 1965.Google Scholar
8. Tang, C. Y., On the Frattini subgroups of generalized free products with cyclic amalgamations, Canad. Math. Bull., 15 (1972), 569-573.Google Scholar
9. Tang, C. Y., On the Frattini subgroups of certain generalized free products of groups, Proc. Amer. Math. Soc., 37 (1973), 63-68.Google Scholar
1. Whittemore, A., On the Frattini subgroup, Trans. Amer. Math. Soc., 141 (1969), 323-333.Google Scholar