Uncancellative factorizations of Baer-local formations.
A function \(f\) which associates with every elementary group \(H\) some (probably empty) formation \(f(H)\) is called a Baer function if \(f(A)=f(B)\) for any two elementary groups \(A\) and \(B\) with \(C(A)=C(B)\). The symbol \(C(A)\) denotes the class of all simple groups \(J\) such that \(J\) is isomorphic to some composition factor of \(A\). A non-empty formation \(\mathfrak F\) is a Baer-local formation, if \({\mathfrak F}=\{G:\) either \(G=1\), or \(G\neq 1\) and \(G/C_G(H/K)\in f(H/K)\) for each chief factor \(H/K\) of \(G\}\), where \(f\) is some Baer-function. Let \(\mathfrak X\) be a set of groups. Remind that \(\text{form\,}{\mathfrak X}\) is the formation generated by \(\mathfrak X\) and \(C({\mathfrak X})\) is the class of all simple groups \(A\) such that \(A\cong H/K\) for some composition factor \(H/K\) of some group \(G\in{\mathfrak X}\). The main result of this paper is the following Theorem. Let \({\mathfrak F}={\mathfrak M}_1{\mathfrak M}_2\cdots{\mathfrak M}_t\) be an uncancellative factorization of a Baer-local formation \(\mathfrak F\) and \({\mathfrak H}={\mathfrak M}_2\cdots{\mathfrak M}_t\). Then \(\mathfrak F\) is a one-generated formation if and only if the following conditions are satisfied: (1) \({\mathfrak M}_1\) is a metanilpotent one-generated local formation and \(C({\mathfrak H})\cap{\mathfrak A}\subseteq{\mathfrak M}_1\); (2) If the formation \({\mathfrak M}_1\) is not primary, then \(\mathfrak H\) is a one-generated formation; besides, if \({\mathfrak M}_1\) is not nilpotent, then \(\mathfrak H\) is Abelian; (3) For all groups \(A\in{\mathfrak M}_1\) and \(B\in{\mathfrak H}\), the groups \(A/F(A)\) and \(B\) have coprime exponents; (4) If \({\mathfrak M}_1={\mathfrak N}_p\) for some prime \(p\), then \({\mathfrak H}/O_p({\mathfrak H})=\text{form}(G/O_p(G):G\in{\mathfrak H})\) is a one-generated formation; (5) \(t\leq3\) and if \(t=3\), then \({\mathfrak M}_1\) and \({\mathfrak M}_2\) are nilpotent, \({\mathfrak M}_3\) is Abelian and the exponents of \({\mathfrak M}_2\) and \({\mathfrak M}_3\) are coprime. -- The theorem gives an answer to Problem~3.5.21 proposed by \textit{A. N. Skiba} in his monograph [see Algebra of formations, Belaruskaya Navuka (1997; Zbl 0926.20014)].
scientific article; zbMATH DE number 1985879