The Hughes subgroup

Robert Bryce

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1994)

  • Volume: 5, Issue: 4, page 283-288
  • ISSN: 1120-6330

Abstract

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Let G be a group and p a prime. The subgroup generated by the elements of order different from p is called the Hughes subgroup for exponent p . Hughes [3] made the following conjecture: if H p ⁢ G is non-trivial, its index in G is at most p . There are many articles that treat this problem. In the present Note we examine those of Strauss and Szekeres [9], which treats the case p = 3 and G arbitrary, and that of Hogan and Kappe [2] concerning the case when G is metabelian, and p arbitrary. A common proof is given for the two cases and a possible lacuna in the first is filled.

How to cite

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Bryce, Robert. "The Hughes subgroup." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 5.4 (1994): 283-288. <http://eudml.org/doc/244276>.

@article{Bryce1994,
abstract = {Let \( G \) be a group and \( p \) a prime. The subgroup generated by the elements of order different from \( p \) is called the Hughes subgroup for exponent \( p \). Hughes [3] made the following conjecture: if \( H\_\{p\}(G) \) is non-trivial, its index in \( G \) is at most \( p \). There are many articles that treat this problem. In the present Note we examine those of Strauss and Szekeres [9], which treats the case \( p = 3 \) and \( G \) arbitrary, and that of Hogan and Kappe [2] concerning the case when \( G \) is metabelian, and \( p \) arbitrary. A common proof is given for the two cases and a possible lacuna in the first is filled.},
author = {Bryce, Robert},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Infinite groups; Hughes subgroup; Metabelian groups; Hughes -conjecture; metabelian groups; associated Lie rings; multilinear identities; free groups of exponent ; 2-Engel Lie algebras; Hughes conjecture; 2-generator 5-groups},
language = {eng},
month = {12},
number = {4},
pages = {283-288},
publisher = {Accademia Nazionale dei Lincei},
title = {The Hughes subgroup},
url = {http://eudml.org/doc/244276},
volume = {5},
year = {1994},
}

TY - JOUR
AU - Bryce, Robert
TI - The Hughes subgroup
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1994/12//
PB - Accademia Nazionale dei Lincei
VL - 5
IS - 4
SP - 283
EP - 288
AB - Let \( G \) be a group and \( p \) a prime. The subgroup generated by the elements of order different from \( p \) is called the Hughes subgroup for exponent \( p \). Hughes [3] made the following conjecture: if \( H_{p}(G) \) is non-trivial, its index in \( G \) is at most \( p \). There are many articles that treat this problem. In the present Note we examine those of Strauss and Szekeres [9], which treats the case \( p = 3 \) and \( G \) arbitrary, and that of Hogan and Kappe [2] concerning the case when \( G \) is metabelian, and \( p \) arbitrary. A common proof is given for the two cases and a possible lacuna in the first is filled.
LA - eng
KW - Infinite groups; Hughes subgroup; Metabelian groups; Hughes -conjecture; metabelian groups; associated Lie rings; multilinear identities; free groups of exponent ; 2-Engel Lie algebras; Hughes conjecture; 2-generator 5-groups
UR - http://eudml.org/doc/244276
ER -

References

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  1. HALL, P., Some sufficient conditions for a group to be nilpotent. Illinois J. Math., 2, 1958, 787-801. Zbl0084.25602MR105441
  2. HOGAN, G. T. - KAPPE, W. P., On the H p -problem for finite p -groups. Proc. Amer. Math. Soc., 20, 1969, 450-454. Zbl0197.02101MR238952
  3. HUGHES, D. R., Research Problem No. 3. Bull. Amer. Math. Soc., 63, 1957, 209. 
  4. HUGHES, D. R. - THOMPSON, J. G., The H p -problem and the structure of H p -groups. Pac. J. Math., 9, 1959, 1097-1101. Zbl0098.25201MR108532
  5. KHUKHRO, E. J., On a connection between Hughes' conjecture and relations in finite groups of prime exponent. Math. USSR Sbornik, 44, 1983, 227-237. Zbl0501.20016
  6. KHUKHRO, E. J., On Hughes problem for finite p -groups. Algebra and Logic, 26, 1987, 136-145. Zbl0658.20015MR985841
  7. LEVI, F. W., Groups in which the commutator relation satisfies certain algebraic conditions. J. Indian Math. Soc., New Ser., 6, 1942, 87-97. Zbl0061.02606MR7417
  8. MACDONALD, I. D., Solution of the Hughes problem for finite p -groups of class 2 ⁢ p - 2 . Proc. Amer. Math. Soc., 27, 1971, 39-42. Zbl0211.04402MR271230
  9. STRAUSS, E. - SZEKERES, G., On a problem of D. R. Hughes. Proc. Amer. Math. Soc., 9, 1958, 157-158. Zbl0081.25605MR93549
  10. WALL, G. E., On Hughes' H p problem. Proceedings of the International Conference on Theory of Groups. Austral. Nat. Univ. Canberra, August 1965, 357-362. Gordon and Breach Science Publishers, Inc. 1967. Zbl0189.31701MR219607
  11. ZAPPA, G., Contributo allo studio del problema di Hughes sui gruppi. Ann. Mat. Pura Appl., (4), 57, 1962, 211-219. Zbl0105.02302MR137761

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