A Gaschütz-Lubeseder type theorem in a class of locally finite groups
The authors develop a Gaschütz-Lubeseder type theorem for a \(c\overline{\mathcal L}\)-formation \(\mathcal F\), \(c\overline{\mathcal L}\) denoting the class of all radical locally finite groups satisfying min-\(p\) for all primes \(p\). Define a semidirect product \(G=[D]M\) to be semiprimitive if \(M\) is a finite soluble group with a trivial core and \(D\) is a divisible Abelian group such that each proper \(M\)-invariant subgroup of \(D\) is finite. The authors relate these two concepts by proving that (a) if \(M\) is a maximal subgroup of \(G\), then \(G/\text{Core}_G(M)\) is a finite primitive soluble group and (b) if \(M\) is not maximal in \(G\), then \(G/\text{Core}_G(M)\) is a semiprimitive group. Next, the properties imposed on the formations required for a local definition are carefully examined. The relationship between a saturated \(c\overline{\mathcal L}\)-formation \(\mathcal F\) and a semiprimitive group \(G\) is proven to be that \(G\in{\mathcal F}\) if and only if \(G\) is the union of an ascending chain of finite \(\mathcal F\)-subgroups. A \(c\overline{\mathcal L}\)-formation \(\mathcal F\) is said to be \(E_\mu\)-closed, \(\mu(G)\) the intersection of the major subgroups, if \(\mathcal F\) satisfies these properties: (a) A \(c\overline{\mathcal L}\)-group \(G\in{\mathcal F}\) if and only if \(G/\mu(G)\in{\mathcal F}\); (b) A semiprimitive group \(G\) is an \(\mathcal F\)-group if and only if it is the union of an ascending chain \(\{G_i:i\in \mathbb{N}\}\) of finite \(\mathcal F\)-groups. The principal result: A \(c\overline{\mathcal L}\)-formation \(\mathcal F\) is \(E_\mu\)-closed if and only if \(\mathcal F\) is a saturated \(c\overline{\mathcal L}\)-formation.
scientific article; zbMATH DE number 1400108
scientific article