On \(\sigma\)-solubility criteria for finite groups (Q6617383)

Let \(G\) be a finite group. A subgroup \(B \leq G\) is supplemented in \(G\) if there exists a subgroup \(A \leq G\) such that \(G = AB\); in this case, \(A\) is a supplement of \(B\) in \(G\).\N\NLet \(\sigma =\{\sigma_{i} \mid i \in I \}\) be a partition of the set of all primes \(\mathbb{P}\). A group \(G\) is \(\sigma\)-primary if \(G\) is a \(\sigma_{i}\)-group for some \(i \in I\), \(G\) is \(\sigma\)-soluble if every chief factor of \(G\) is \(\sigma\)-primary and \(G\) is \(\sigma\)-nilpotent if it is a direct product of \(\sigma\)-primary groups.\N\NIn [\textit{W. Guo} et al., Isr. J. Math. 138, 125--138 (2003; Zbl 1050.20009)], it was shown that a group \(G\) is nilpotent (respectively, supersoluble) if every maximal subgroup of every Sylow subgroup of \(G\) has a nilpotent (respectively, supersoluble) supplement in \(G\). Subsequently, this result was extended to the solubility of groups in which every maximal subgroup of every Sylow subgroup of \(G\) has a soluble supplement (see, in particular, [\textit{A. M. Liu} et al., J. Algebra 585, 280--293 (2021; Zbl 1534.20021)]).\N\NThe main theorems of the paper under review are the following generalizations of the results given above.\N\NTheorem 1.3: Let \(p\) be a prime dividing the order of a group \(G\) and \(P \in \mathrm{Syl}_{p}(G)\). Assume that \(p \not \in \{ 7, 13 \}\). If every maximal subgroup of \(P\) has a soluble supplement in \(G\), then \(G\) is soluble.\N\NAnother interesting result is Theorem 1.6: Assume that \(p\) and \(q\) are two different primes dividing the order of a group \(G\) and \(P\in \mathrm{Syl}_{p}(G)\), \(Q \in \mathrm{Syl}_{q}(G)\). If every maximal subgroup of \(P\) and \(Q\) has a \(\sigma\)-soluble supplement in \(G\), then \(G\) is \(\sigma\)-soluble.\N\NThe authors observe that the hypothesis \(p \not \in \{ 7, 13 \}\) in Theorem 1.3 cannot be removed since \(G=\mathrm{PSL}_{3}(2) \simeq \mathrm{PSL}_{2}(7)\) has a Sylow subgroup of order \(7\) with a supplement \(H \simeq S_{4}\) and \(G=\mathrm{PSL}_{3}(3)\) has a Sylow subgroup of order \(13\) with a supplement \(H \simeq (C_{3} \times C_{3}) \rtimes \mathrm{GL}_{2}(3)\). The reviewer confesses that she did not understand how the authors treat the case \(p=5\) as \(G=\mathrm{PSL}_{2}(5) \simeq A_{5}\) has a Sylow subgroup of order \(5\) with supplement \(H \simeq A_{4}\).