On the solvability of finite groups (Q5903769)

In earlier papers the author has shown that normality of certain subgroups insures solvability or supersolvability. In this paper he shows that normality can be replaced by quasinormality. Let G be a finite group and define \(A_ 1=\{H\leq G:\) H has prime order or is cyclic of order \(4\}\), \(A_ 2=\{H\leq G:\) H has order 2p, p an odd prime\(\}\) and \(A_ 3=\{H\leq G:\) H has order 2pq, p and q primes not necessarily odd or distinct\(\}\). Theorem 3.1. If each element of \(A_ 1\) is quasinormal in G then G is supersolvable. Theorem 3.2. If G has even order and each element of \(A_ 2\) is quasinormal in G then G is solvable. Theorem 3.3. If G has even order and each element of \(A_ 3\) is quasinormal in G then G is solable. (Note: The proof of Theorem 3.3. does not seem to allow \(A_ 3\) to contain a subgroup of order \(2p^ 2\), p an odd prime.)