Determining hypercentral Hall subgroups in finite groups (Q6607384)

Let \(G\) be a finite group, \(x \in G\) and let \(|x^{G}|=|G:C_{G}(x)|\) be the size of the conjugacy class of \(x\) in \(G\). In order to study the influence of conjugate class sizes on the structure of \(G\), \textit{J. Cossey} et al. [Bull. Lond. Math. Soc. 24, No. 3, 267--270 (1992; Zbl 0797.20021)] have introduced \(w^{G}\), the class-size frequency function of \(G\). Let \(\pi \subseteq \pi(G)\) and let \(G_{\pi}\) be the set of \(\pi\)-elements of \(G\). In the paper under review, the author introduces, in a similar way to [loc. cit.], the \(\pi\)-class size frequency function of \(G\), defined by\N\[\Nw_{\pi}^{G}: \mathbb{N} \rightarrow \mathbb{N} \;\; n \mapsto \frac{1}{n} \cdot \big | \{g \in G_{\pi} \mid |g^{G}| \} \big |.\N\]\NThe main result is Theorem A: Let \(G\) and \(H\) be finite groups, and suppose that \(G\) has a hypercentral Hall \(\pi\)-subgroup for certain set of primes \(\pi\). If the conjugacy class sizes of the \(\pi\)-elements of \(G\) joint with their multiplicities are the same as those of \(H\), then \(H\) has a hypercentral Hall \(\pi\)-subgroup. As a corollary, the author proves that, if \(G\) and \(H\) are groups, \(G\) is nilpotent and for a given prime \(p\), the class sizes of the \(p\)-regular elements of \(G\) and their multiplicities coincide with those of \(H\), then \(H\) is nilpotent.