×

Order evaluation of products of subsets in finite groups and its applications. II. (English) Zbl 0895.20022

Let \(G\) be a finite group and \(A\) be a subset of \(G\). Then \(A\) is called a normal subset of \(G\) if \(A\) is invariant under conjugation. In part I [J. Algebra 182, No. 3, 577-603 (1996; Zbl 0862.20017)]Z. Arad, E. Fisman and M. Muzychuk proved that if \(G\) is a nonabelian finite simple group and \(A\) is a normal subset of \(G\) such that \(| A|>1\), then \(| AB|\geq| A|+| B|-1\) holds for all subsets \(B\) of \(G\). In the present paper the authors improve the bound given above and the following main theorem is proved. Let \(G\) be a nonabelian finite simple group and let \(l\) be the minimal cardinality of a nontrivial conjugacy class of \(G\). If \(A\) is a normal subset of \(G\) such that \(1<| A|\leq| G|/4\), then for any subset \(B\) of \(G\) for which \(| B|\geq 2\) and \(| AB|\leq| G|-2\) the inequality \(| AB|\geq| A|+| B|+{l-18\over 12}\) holds. In particular the authors prove that under the above assumptions the inequality \(| AB|\geq| A|+| B|+3\) holds as well. The results of this paper have interesting consequences concerning the product of conjugacy classes in finite groups.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D05 Finite simple groups and their classification
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)

Citations:

Zbl 0862.20017
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Zvi Arad and Harvey I. Blau, On table algebras and applications to finite group theory, J. Algebra 138 (1991), no. 1, 137 – 185. · Zbl 0790.20015 · doi:10.1016/0021-8693(91)90195-E
[2] Z.Arad, E.Fisman, M.Muzychuk. Order evaluation of products of subsets in finite groups and its applications.I, J. Algebra 182 (1996), 577-603. CMP 96:15 · Zbl 0862.20017
[3] John D. Dixon and Brian Mortimer, The primitive permutation groups of degree less than 1000, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 2, 213 – 238. · Zbl 0646.20003 · doi:10.1017/S0305004100064793
[4] Walter Feit, Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, 1967. · Zbl 0228.20019
[5] Walter Feit and John G. Thompson, Finite groups which contain a self-centralizing subgroup of order 3., Nagoya Math. J. 21 (1962), 185 – 197. · Zbl 0114.25602
[6] Bertram Huppert and Norman Blackburn, Finite groups. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 242, Springer-Verlag, Berlin-New York, 1982. AMD, 44. Bertram Huppert and Norman Blackburn, Finite groups. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 243, Springer-Verlag, Berlin-New York, 1982.
[7] I. M. Isaacs and Ilan Zisser, Squares of characters with few irreducible constituents in finite groups, Arch. Math. (Basel) 63 (1994), no. 3, 197 – 207. · Zbl 0873.20005 · doi:10.1007/BF01189820
[8] B. A. Pogorelov, Primitive groups of permutations of small degrees. I, Algebra i Logika 19 (1980), no. 3, 348 – 379, 383 (Russian). B. A. Pogorelov, Primitive groups of permutations of small degrees. II, Algebra i Logika 19 (1980), no. 4, 423 – 457, 503 (Russian).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.