On a lower bound of commutativity degree.
There is a growing interest in the last years in the so-called ``commutativity degree'' of a finite group \(G\): this is defined as the number \(d(G)=\tfrac 1{|G|^2}|\{(x,y)\in G^2\mid [x,y]=1\}|\). The authors of this paper are devoting many researches in the last years to the topic, generalizing a series of recent contributions in the literature. It is a common opinion that we may investigate \(d(G)\) either with combinatorial techniques or by using character theory, as shown by \textit{W. H. Gustafson}, [Am. Math. Mon. 80, 1031-1034 (1973; Zbl 0276.60013)], \textit{R. M. Guralnick, G. R. Robinson}, [J. Algebra 300, No. 2, 509-528 (2006; Zbl 1100.20045), Addendum ibid. 319, No. 4, 1822 (2008; Zbl 1176.20067)], \textit{A. Erfanian, R. Rezaei, P. Lescot}, [Commun. Algebra 35, No. 12, 4183-4197 (2007; Zbl 1152.20023)], \textit{M. R. R. Moghaddam, A. R. Salemkar, K. Chiti}, [Algebra Colloq. 12, No. 2, 255-261 (2005; Zbl 1071.20027)], \textit{P. Lescot}, [J. Algebra 177, No. 3, 847-869 (1995; Zbl 0841.20026)], \textit{A. Erfanian, F. Russo}, [Ital. J. Pure Appl. Math. 25, 27-36 (2009; Zbl 1190.20055), Bull. Iran. Math. Soc. 34, No. 2, 27-37 (2008; Zbl 1202.22006)], \textit{R. Rezaei, F. Russo}, [Ischia group theory 2008. Proceedings of the conference in group theory, Naples, Italy 2008. Hackensack, NJ: World Scientific. 233-235 (2009; Zbl 1183.22003)] and in other recent contributions of the same authors. Furthermore, we may treat \(d(G)\) in the infinite case, once we look at the compact groups (see again the above citations). In the present paper, the authors deal with finite groups and provide the lower bound \(d(G)\geq\tfrac{1}{|G'|}\Bigl(1+\tfrac{|G'|-1}{|G:Z(G)|}\Bigr)\) in Theorem 1, which is very general. This allows them to find interesting restrictions in Theorem 2 on the size of the conjugacy classes of \(G\). Applications to the \(p\)-groups can be found in Section 3, where the condition \(G'\subseteq Z(G)\) is treated by means of numerical restrictions on \(d(G)\).
scientific article; zbMATH DE number 5723779
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