Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-06T10:53:30.499Z Has data issue: false hasContentIssue false
  • English
  • Français

B.H. Neumann's Question on Ensuring Commutativity of Finite Groups

Published online by Cambridge University Press:  17 April 2009

A. Abdollahi ,
A. Azad ,
A. Mohammadi Hassanabadi  and
M. Zarrin
A. Abdollahi
Affiliation:
Department of MathematicsUniversity of IsfahanIsfahan 81746-73441Iran e-mail: [email protected]
A. Azad
Affiliation:
Department of MathematicsUniversity of IsfahanIsfahan 81746-73441Iran e-mail: [email protected]
A. Mohammadi Hassanabadi
Affiliation:
Department of MathematicsUniversity of IsfahanIsfahan 81746-73441Iran e-mail: [email protected]
M. Zarrin
Affiliation:
Department of MathematicsUniversity of IsfahanIsfahan 81746-73441Iran e-mail: [email protected]
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.

This paper is an attempt to provide a partial answer to the following question put forward by Bernhard H. Neumann in 2000: “Let G be a finite group of order g and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. What relations between g, m, n guarantee that G is Abelian?” We find an exponential function f(m,n) such that every such group G is Abelian whenever |G| > f(m,n) and this function can be taken to be polynomial if G is not soluble. We give an upper bound in terms of m and n for the solubility length of G, if G is soluble.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 74 , Issue 1 , August 2006 , pp. 121 - 132
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Abdollahi, A., Akbari, S. and Maimani, H.R., ‘Non-commuting graph of a group’, J. Algebra 298 (2006), 468–492.CrossRefGoogle Scholar
[2]Abdollahi, A. and Amiri, S.M. Jafarian, ‘On groups with an irredundant 7-cover’, J. Pure Appl. Algebra (to appear).Google Scholar
[3]Abdollahi, A. and Hassanabadi, A. Mohammadi, ‘Finite groups with certain number of elements pairwise generating a non-nilpotent subgroup’, Bull. Iranian Math. Soc. 30 (2004), 1–20.Google Scholar
[4]Bertram, E.A., ‘Some applications of graph theory to finite groups’, Discrete Math. 44 (1983), 31–43.Google Scholar
[5]Blyth, R.D. and Robinson, D.J.S., ‘Insoluble groups with the rewriting property P8’, J. Pure Appl. Algebra 72 (1991), 251–263.Google Scholar
[6]Bryce, R.A., Fedri, V. and Serena, L., ‘Covering groups with subgroups’, Bull. Austral. Math. Soc. 55 (1997), 469–476.Google Scholar
[7]Endimioni, G., ‘Groupes finis satisfaisant la condition (N, n)’, C. R. Acad. Sci. Paris Ser. I Math. 319 (1994), 1245–1247.Google Scholar
[8]Huppert, B. and Blackburn, N., Finite groups, III (Springer-Verlag, New York, 1982).Google Scholar
[9]Neumann, B.H., ‘A problem of Paul Erdos on groups’, J. Austral. Math. Soc. Ser. A 21 (1976), 467–472.CrossRefGoogle Scholar
[10]Neumann, B.H., ‘Ensuring commutativity of finite groups’, J. Aust. Math. Soc. 71 (2001), 233–234.CrossRefGoogle Scholar
[11]Pyber, L., ‘The number of pairwise non-commuting elements and the index of the centre in a finite group’, J. London Math. Soc. (2) 35 (1987), 287–295.Google Scholar
[12]Segal, D. and Shalev, A., ‘On groups with bounded conjugacy classes’, Quart. J. Math. Oxford Ser. (2) 50 (1999), 505–516.Google Scholar
[13]Thompson, J.G., ‘Nonsolvable finite groups all of whose local subgroups are soluble’, Bull. Amer. Math. Soc. 74 (1968), 383–437.Google Scholar