Subgroup-closed lattice formations.
The paper under review deals with finite groups. Therefore, the word ``group'' will mean ``finite group''. Recall that a class of finite groups that is closed under taking quotients and subdirect products is called a \textit{formation}. If every subgroup of a group in a formation \(\mathfrak F\) is also in \(\mathfrak F\), then \(\mathfrak F\) is called subgroup-closed. Moreover, if the formation \(\mathfrak F\) is closed under Frattini extensions, \(\mathfrak F\) is said to be saturated. If \(\mathfrak F\) is a formation, then every group \(G\) has a smallest normal subgroup \(G^{\mathfrak F}\), called the \textit{\(\mathfrak F\)-residual} of \(G\), for which the corresponding quotient is in \(\mathfrak F\). A subgroup \(H\) of a group \(G\) is said to be \textit{\(\mathfrak F\)-subnormal} (respectively, \textit{K-\(\mathfrak F\)-subnormal}) in \(G\) if either \(H=G\) or there exists a maximal chain of subgroups \(H=H_0\leq H_1\leq\cdots\leq H_n=G\) such that \(H_i^{\mathfrak F}\leq H_{i-1}\) (respectively, \(H_{i-1}\) is a normal subgroup of \(H_i\) or \(H_i^{\mathfrak F}\leq H_{i-1}\)) for \(i\in\{1,\ldots,n\}\). In general, for a subgroup-closed formation \(\mathfrak F\), the set of all \(\mathfrak F\)-subnormal (respectively, K-\(\mathfrak F\)-subnormal) subgroups of a group \(G\) is not a lattice of the subgroup lattice of \(G\). The authors obtain that the following statements are equivalent for a subgroup-closed formation \(\mathfrak F\): 1. The set of all K-\(\mathfrak F\)-subnormal subgroups is a sublattice of the subgroup lattice of every group. 2. The set of all \(\mathfrak F\)-subnormal subgroups is a sublattice of the subgroup lattice of every group. 3. \(\mathfrak F=\mathfrak M\times\mathfrak K\times\mathfrak L\) for some subgroup-closed formations \(\mathfrak M\), \(\mathfrak K\), and \(\mathfrak L\) satisfying the following conditions: (a) \(\pi(\mathfrak M)\cap\pi(\mathfrak K)=\emptyset\), \(\pi(\mathfrak K)\cap\pi(\mathfrak L)=\emptyset\), and \(\pi(\mathfrak M)\cap\pi(\mathfrak L)=\emptyset\); (b) \(\mathfrak M=\mathfrak S_{\pi(\mathfrak M)}\mathfrak M\) is a saturated formation and an \(\mathfrak M^2\)-normal Fitting class; (c) every non-cyclic \(\mathfrak M\)-critical group \(G\) with \(\Phi(G)=1\) is a primitive group of type 2 such that \(G/\mathrm{Soc}(G)\) is a cyclic group of prime power order; (d) there exists a partition \(\{\pi_j\mid j\in J\}\) of \(\pi(\mathfrak K)\) such that \(\mathfrak K=\bigtimes_{j\in J}\mathfrak S_{\pi_j}\) and \(|\pi_j|>1\) for all \(j\in J\); (e) \(\mathfrak L\subseteq\mathfrak N_{\pi(\mathfrak L)}\). This characterisation generalises some known results about subgroup-closed saturated formations or subgroup-closed formations of soluble groups.
scientific article; zbMATH DE number 6493622