Arithmetic of conjugacy of \(p\)-complements.
Let \(G\) be a finite group and \(p\in\pi(G)\). A subgroup \(H\) of \(G\) is called a \(p\)-complement (or a \(p'\)-Hall subgroup) if for any \(p\)-Sylow subgroup \(P\) of \(G\) the equalities \(HP=G\) and \(H\cap P=1\) hold. A well-known theorem of \textit{P. Hall} [Journal L.M.S. 3, 98-105 (1928; JFM 54.0145.01)] asserts that a finite group \(G\) is solvable if and only if \(G\) admits a complement for each \(p\in\pi(G)\). For every prime number \(p\) the author defines the following classes of finite groups admitting a \(p\)-complement: \(\bullet\) \(\mathfrak C(p)\), the class of groups the \(p\)-complements in which are all conjugate; \(\bullet\) \(\mathfrak A(p)\), the class of groups \(G\) such that every two \(p\)-complements in \(G\) are conjugate by an element of \(\Aut(G)\); \(\bullet\) \(\mathfrak I(p)\), the class of groups in which the \(p\)-complements are isomorphic. The following chain of inclusions holds: (*) \(\mathfrak C(p)\subseteq\mathfrak A(p)\subseteq\mathfrak I(p)\). Denote by \(\mathcal{NC}\) the set of prime numbers \(p\) such that \(p^k=(q^r-1)/(q-1)\) for some \(k\in\mathbb N\), where \(q\) is a power of a prime and \(r\) is an odd prime. The main result of this paper is Theorem 1: Let \(p\) be some prime number. Then one of the following statements holds: (a) all inclusions in chain (*) are strict and \(p\in\mathcal{NC}\); (b) all inclusions in chain (*) are equalities and \(p\not\in\mathcal{NC}\). It is interesting to note that \(2,3,5\not\in\mathcal{NC}\) and \(7,11,13\in\mathcal{NC}\); moreover the author proves (Proposition 2), among other things, that if \(p\) is a Mersenne prime greater than \(3\), then \(p\in\mathcal{NC}\).
scientific article; zbMATH DE number 6485742
scientific article