Lattice and group complementability in periodic locally solvable groups (Q1068936)

A subgroup B of a group G is called a lattice complement of a subgroup A if \(G=<A,B>\) and \(A\cap B=1\). The subgroup A is said to be \(G\Phi\)- complemented if there exists a subgroup H of G such that \(G=AH\) and \(A\cap H\subseteq \Phi (G)\), where \(\Phi\) (G) denotes the Frattini subgroup of G. A group G is said to be a \(\Phi_{\omega}C\)-group if the following conditions on a subgroup A of G imply that A is \(G\Phi\)- complemented: A is a p-subgroup of G for some prime p and A contains a Sylow p-subgroup of \(\Phi\) (G); A has a lattice complement B which contains a Sylow p'-subgroup of \(\Phi\) (G). Let G be a periodic locally solvable groups which is locally normal or has finite Sylow p-subgroups for all primes p. Theorem: if G is a \(\Phi_{\omega}C\)-group then G is locally supersolvable. There are some further related results.