Groups of \(p\)-central type (Q6139838)

Let \(G\) be a finite group, then for every irreducible complex character \(\chi \in \mathrm{Irr}(G)\) it is easy to show that \(\chi(1)^{2} \leq |G: Z(G)|\). If equality holds for some \(\chi \in \mathrm{Irr}(G)\), then \(G\) is said to be of central type. Using the classification of finite simple groups, \textit{R. B. Howlett} and \textit{I. M. Isaacs} [Math. Z. 179, 555--569 (1982; Zbl 0511.20002)] proved that all groups of central type are solvable. A corresponding theorem for \(p\)-Brauer characters was proved by \textit{G. Navarro} et al. [Adv. Math. 257, 248--265 (2014; Zbl 1322.20009)] under the assumption that \(p \not = 5\). In the paper under review, the author shows that the restriction \(p \not =5\) can be removed thus obtaining that every group of \(p\)-central type is solvable. \textit{D. M. Gagola} [Pac. J. Math. 55 (1974), 107--126 (1973; Zbl 0283.20003)] has shown that every solvable group can be embedded into \(G/Z(G)\) for some group \(G\) of central type. The author extends Gagola's result by proving Theorem 2: Let \(G\) be a solvable group and \(p\) be a prime. Then there exists a group \(H\) of \(p\)-central type such that \(G\) is isomorphic to a subgroup of \(H/Z(H)\).