Blocks of small defect in alternating groups and squares of Brauer character degrees (Q1676217)

The nilpotency of a finite group was characterized by means of a character theoretic condition by \textit{S. M. Gagola jun.} and \textit{M. L. Lewis} [Commun. Algebra 27, No. 3, 1053--1056 (1999; Zbl 0929.20010)]. They established that a finite group \(G\) is nilpotent if and only if \(\chi(1)^2\) divides \(|G : \text{ker}(\chi)|\) for all \(\chi\in \text{Irr}(G)\). In the paper under review, the authors study a similar divisibility condition for Brauer characters. The main result, Theorem 1, asserts that if \(p\) is a prime and \(G\) be a finite group, then \(\varphi(1)^2\) divides \(|G : \text{ker}(\varphi)|\) for all \(\varphi \in\text{IBr}(G)\) if and only if \(G\) has a normal Sylow \(p\)-subgroup \(P\) and \(G/P\) is nilpotent. The greatest difficulty of the proof of Theorem 1 is to prove that if the divisibility condition holds then the Sylow \(p\)-subgroup is normal in \(G\). The \(p\)-solvable and non-\(p\)-solvable cases are considered separately. For the \(p\)-solvable case, the authors make use, among other techniques, of the B\(_\pi\)-character theory introduced by \textit{I. M. Isaacs} [J. Algebra 86, 98--128 (1984; Zbl 0526.20006)]. In the non-\(p\)-solvable case, the authors manage to reduce the question to whether a certain non-abelian composition factor \(S\) of \(G\) has a defect zero character or not. The former case quickly leads to a contradiction, whereas the second case yields to \(p = 2\) or \(3\) and \(S\) is isomorphic either to certain sporadic simple groups or an alternating group Alt(\(n\)) for \( n \geq 7\). The sporadic groups are handled using GAP and, for the alternating groups, the authors obtain first the following result. Theorem 2. Let \(n\geq 5\) be an integer, let \(p = 2\) or \(3\) and let \(p^a = |\text{Alt}(n)|_p\). Then, \(\text{Alt}(n)\) has a \(p\)-block \(B\) of defect \(d = d(B)\) such that \(d \leq (a - 1)/2\) for \( p = 3\), and \(d = (a - 2)/2\) for \(p = 2\) with the following exceptions: \( n = 7\) if \(p = 3\); and \(n = 7, 9, 11, 13, 22, 24\) and \(26\) if \(p = 2\). Theorem 2 is related to a question posed by \textit{A. Espuelas} and \textit{G. Navarro} [Proc. Am. Math. Soc. 114, No. 4, 881--885 (1992; Zbl 0747.20005)], where they asked whether a finite group \( G\) satisfying \(\mathbf O_p(G) = 1\) and \(|G|_p = p^a\) with \(p \geq 5\) contains a block of defect less than \([a/2]\).