Finite groups without elements of order \(10\): the case of solvable or almost simple groups (Q6575417)

Let \(G\) be a finite group, \(\pi(G)\) the set of all prime divisors of \(|G|\) and let \(\varpi(G)\) be set of orders of the elements of \(G\).\N\NSolvable groups in which \(\varpi(G)\) contains only prime powers were classified independently by \textit{G. Zacher} [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 20, 171--174 (1956; Zbl 0075.23906)] and \textit{G. Higman} [J. Lond. Math. Soc. 32, 335--342 (1957; Zbl 0079.03204)]. \textit{M. Suzuki} [Ann. Math. (2) 75, 105--145 (1962; Zbl 0106.24702)] found the simple groups with this property and then \textit{R. Brandl} [Boll. Unione Mat. Ital., V. Ser., A 18, 491--493 (1981; Zbl 0473.20013)] completed the classification of these groups. The non-abelian finite simple groups \(G\) with \(3 \in \pi(G)\) and \(6 \not \in \varpi(G)\) were determined in three independent articles by \textit{N. D. Podufalov} [Algebra Logic 16, 133--135 (1978; Zbl 0411.20011)], \textit{L. F. Fletcher} et al. [Q. J. Math., Oxf. II. Ser. 28, 143--154 (1977; Zbl 0363.20018)] and \textit{L. M. Gordon} [Bull. Aust. Math. Soc. 17, 235--246 (1977; Zbl 0366.20007)].\N\NThe paper under review is devoted to the study of groups \(G\) such that \(p \in \pi(G)\), \(p>3\) and \(2p \not \in \varpi(G)\). In Theorem 1, the authors classify the almost simple groups with \(5 \in \pi(G)\) and \(10 \not \in \varpi(G)\). In the reviewer's opinion, the main result of this paper is Theorem 2: \N\NGiven a prime \(p > 3\), if \(G\) is a finite solvable group without elements of order \(2p\) and \(2p\) dividing \(|G|\) then one of the following holds: \N\begin{itemize}\N\item[(1)] \(G/O(G)\) is isomorphic to a cyclic group or a (generalized) quaternion \(2\)-group, \(\mathrm{SL}_{2}(3)\) or \(\mathrm{SL}_{2}(3).2\), a Sylow \(p\)-subgroup of \(O(G)\) is abelian and \(O(G)\) is of \(p\)-length \(1\); \N\item[(2)] \(G/O_{p'}(G)\) is a cyclic \(p\)-group or a Frobenius group with cyclic core of order \(|G|_{p}\) and cyclic complement of order dividing \(p-1\), the degree of nilpotence of a Sylow \(2\)-subgroup of \(O_{p'}(G)\) is at most \((p^{2}-1)/4\) (and this estimate is sharp for \(p=5\)) and \(O_{p'}(G)\) has \(2\)-length at most \(1\).\N\end{itemize}\N\NThe reviewer points out that \textit{G. Higman} [J. Lond. Math. Soc. 32, 321--334 (1957; Zbl 0079.03203)] proved that if a nilpotent group $S$ admits an automorphism $\alpha$ of prime order $p$, then the nilpotency class of $S$ is bounded by a function $k(p)$ of $p$. It is easy to see that $k(2)=1$ and that $k(3)=2$ and Higman showed that $k(5)=6$. Based on these data he conjectured that $k(p)=(p^{2}-1)/4$ and exhibited examples where that limit is actually reached (for a discussion see [\textit{E.~I. Khukhro}, Nilpotent groups and their automorphisms. De Gruyter Expositions in Mathematics 8. Berlin: Walter de Gruyter (1993; Zbl 0795.20018), \S 4.3]). It was also proved by \textit{B.~Scimemi} [unpublished] using a computer and reproved by \textit{I.~Hughes} [C. R. Math. Acad. Sci., Soc. R. Can. 7, 61--66 (1985; Zbl 0563.20024)] with hand calculations, that $k(7)=12$. Higman's conjecture is still open for $p>7$ and the only known bounds for $k(p)$ are more than exponential. So Lemma~5, as it is stated in the paper, is only proven for $p \leq 7$. Since the paper concerns the case $p=5$ this is not a serious error (but this requires a slight modification of (2) in Theorem~2).