On chief factors of finite groups.
Let \(G\) be a finite group and \(\mathcal F\) the set of its maximal subgroups. The following notations are adopted: \(\mathcal F_n=\{M\in\mathcal F:M\) non-nilpotent\}, \(\mathcal F_c=\{M\in\mathcal F:[G:M]\) is composite\}, \(\mathcal F^p=\{M\in\mathcal F:N_G(P)\leq M\) for some Sylow \(p\)-subgroup \(P\) of \(G\}\), \(\mathcal F^{op}=\bigcup_{p\in \pi(G)\setminus\{2\}}\mathcal F^p\), \(\mathcal F^{ocn}=\mathcal F^{op}\cap\mathcal F_c\cap\mathcal F_n\). Theorem 1: If \(H/K\) is a chief factor of \(G\), then each of the following statements implies the other four: (i) \(H/K\) is soluble; (ii) every maximal subgroup of \(G\) either covers or avoids \(H/K\); (iii) every maximal subgroup of \(G\) in \(\mathcal F^{ocn}\) covers or avoids \(H/K\); (iv) every Hall subgroup of \(G\) covers or avoids \(H/K\); (v) there exists a prime \(p\in\pi(H/K)\) and a Sylow \(p\)-subgroup \(P\) of \(G\) such that \(P\) covers or avoids \(H/K\). A subgroup \(T\) of \(G\) is called a 2-maximal subgroup of \(G\) if \(T\) is a maximal subgroup of \(S\), for some maximal subgroup \(S\) of \(G\). Theorem 2: If \(H/K\) is a chief factor of \(G\), then \(H/K\) is soluble if one of the following statements is true: (i) every 2-maximal subgroup of \(G\) covers or avoids \(H/K\); (ii) there exists a maximal subgroup \(M\) of \(G\) with \([G:M]\) having at most two distinct prime divisors such that \(M\) covers or avoids \(H/K\); (iii) there exists a soluble maximal subgroup \(M\) of \(G\) such that \(M\) covers or avoids \(H/K\); (iv) every maximal subgroup of every Sylow \(p\)-subgroup of \(G\) covers or avoids \(H/K\), where \(p=\min\pi(H/K)\). Theorem 3: Suppose there exists a chief series \(\mathcal H\) of \(G\) passing through the generalized Fitting subgroup \(F^*\) of \(G\) such that every maximal subgroup of every Sylow subgroup of \(F^*\) either covers or avoids each chief factor of \(G\) in \(\mathcal H\). Then \(G\) is soluble. A subgroup \(L\) of \(G\) is called a pure 2-maximal subgroup of \(G\) if \(L\) is a maximal element in the set of 2-maximal subgroups of \(G\). Theorem 4: Let \(H/K\) be a chief factor of \(G\). If some soluble pure 2-maximal subgroup of \(G\) avoids \(H/K\), then \(H/K\) is soluble.
scientific article; zbMATH DE number 5167580