Nonpronormal subgroups of odd index in finite simple linear and unitary groups (Q6597562)

Let \(G\) be a finite group, a subgroup \(H \leq G\) is pronormal in \(G\) if, for every \(g \in G\), the subgroups \(H\) and \(H^{g}\) are conjugate in \(\langle H, H^{g} \rangle\). Well-known examples of pronormal subgroups are normal subgroups, maximal subgroups, Sylow subgroups and Hall subgroups of solvable groups.\N\NThe third author and \textit{E. P. Vdovin}, in [Sib. Math. J. 53, No. 3, 419--430 (2012; Zbl 1275.20008)], proved that all Hall subgroups are pronormal in finite simple groups and conjectured that all subgroups of odd index are pronormal in simple groups. The conjecture was refuted quite soon by \textit{A. S. Kondrat'ev} and the second and third authors, in [Proc. Steklov Inst. Math. 296, Suppl. 1, S145--S150 (2017; Zbl 1371.20009)], and instead a program was developed for classifying finite simple groups in which all subgroups of odd index are pronormal.\N\NFor two natural numbers \(a\) and \(b\) with binary expansions \(a=\sum_{i=0}^{\infty} \alpha_{i} 2^{i}\) and \(b=\sum_{i=0}^{\infty} \beta_{i} 2^{i}\) where \(\alpha_{i}, \beta_{i}\in \{0, 1\}\) and almost all \(\alpha_{i}\) and \(\beta_{i}\) are zero, define \(a \preceq b\) if and only if \(\alpha_{i} \leq \beta_{i}\) for all \(i\).\N\NThe main results of the paper under review are as follows.\N\NTheorem 1: Let \(G=\mathrm{PSL}^{\epsilon}_{n}(q)\), where \(n \geq 3\), \(\epsilon \in \{ +, -\}\), and \(q\) is a power of an odd prime. Assume that \(q \equiv \epsilon 1 \mod 4\) and there exists a natural number \(m\) such that \(m \preceq n\) and the number \((q-\epsilon 1,m)\) is not a power of two. Then \(G\) contains a nonpronormal subgroup of odd index.\N\NTheorem 2: Let \(G=\mathrm{PSL}_{n}(q)\), where \(n\geq 3\), \(q\) is odd, \(q=q_{0}^{r}\) for an odd prime \(r\) not dividing \(q_{0}-1\), and there exists a number \(m \preceq n\) such that \(\big (\frac{q-1}{q_{0}-1},m \big) \not =1\). Then the group \(G\) contains a nonpronormal subgroup of odd index.