Arithmetical conditions on the length of the conjugacy classes of a finite group
In this paper groups are finite. First, using new tools, another proof of Theorem 2 of \textit{E. A. Bertram, M. Herzog} and \textit{A. Mann} [Bull. Lond. Math. Soc. 22, No. 6, 569-575 (1990; Zbl 0743.20017)] is given (Theorem 4). Definitions: If \(G\) is a group, then \(\rho' (G)\) is the set of all primes dividing the cardinality of some conjugacy class of \(G\); \(\Gamma' (G)\) is the graph with vertex set \(\rho' (G)\) whose element pairs \((p,q)\) are connected by an edge if \(pq\) divides the cardinality of some conjugacy class in \(G\). Theorem 13: Let \(G\) be a soluble group, and let \(p\) and \(q\) be different elements in \(\rho' (G)\) not joined by an edge in \(\Gamma'(G)\), then both the Sylow \(p\)- and the Sylow \(q\)- subgroups of \(G\) are abelian (and thus both the \(p\)-length and the \(q\)- length of \(G\) are 1) and \(G\) has either a normal \(p\)- or a normal \(q\)- complement. Further results about the graph \(\Gamma' (G)\) are derived (Theorems 16 and 17).
scientific article; zbMATH DE number 783888