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ON NONNILPOTENT SUBSETS IN GENERAL LINEAR GROUPS

Part of: Representation theory of groups

Published online by Cambridge University Press:  01 April 2011

AZIZOLLAH AZAD
AZIZOLLAH AZAD*
Affiliation:
Department of Mathematics, Faculty of Sciences, Arak University, Arak 38156-8-8349, Iran (email: [email protected])
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Abstract

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Let G be a group. A subset X of G is said to be nonnilpotent if for any two distinct elements x and y in X, 〈x,y〉 is a nonnilpotent subgroup of G. If, for any other nonnilpotent subset X′ in G, ∣X∣≥∣X′ ∣, then X is said to be a maximal nonnilpotent subset and the cardinality of this subset is denoted by ω(𝒩G) . Using nilpotent nilpotentizers we find a lower bound for the cardinality of a maximal nonnilpotent subset of a finite group and apply this to the general linear group GL (n,q) . For all prime powers q we determine the cardinality of a maximal nonnilpotent subset of the projective special linear group PSL (2,q) , and we characterize all nonabelian finite simple groups G with ω(𝒩G)≤57 .

Keywords

general linear groupnonnilpotent set

MSC classification

Secondary: 20D60: Arithmetic and combinatorial problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 3 , June 2011 , pp. 369 - 375
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Abdollahi, A., Akbari, A. and Maimani, H. R., ‘Noncommuting graph of a group’, J. Algbera 298 (2006), 468–492.CrossRefGoogle Scholar
[2]Abdollahi, A. and Mohammadi Hassanabadi, A., ‘Finite groups with a certain number of elements pairwise generating a nonnilpotent subgroup’, Bull. Iranian Math. Soc. 30(2) (2004), 1–20, 99–100.Google Scholar
[3]Blyth, R. D. and Robinson, D. J. S., ‘Semisimple groups with the rewriting property Q 5’, Comm. Algebra 23(6) (1995), 2171–2180.CrossRefGoogle Scholar
[4]Endimioni, G., ‘Groups finis satisfaisant la condition (𝒩,n)’, C. R. Acad. Sci. Paris Ser. I 319 (1994), 1245–1247.Google Scholar
[5]Huppert, B., Endliche Gruppen. I (Springer, Berlin, 1967).CrossRefGoogle Scholar
[6]Huppert, B. and Blackburn, N., Finite Groups. III (Springer, Berlin, 1982).CrossRefGoogle Scholar
[7]Tomkinson, M. J., ‘Hypercentre-by-finite groups’, Publ. Math. Debrecen 40 (1992), 313–321.CrossRefGoogle Scholar