The Influence on a Finite Group of Its Proper Abnormal Structure

On a class of generalised Schmidt groups

“…Rose [12] studied the effects of replacing maximal subgroups by nonnormal (or abnormal) maximal subgroups in the hypothesis of Schmidt's result, and the following fact is established:…”

“…The results presented here spring from the classical results of Schmidt [13] about the structure of the minimal non-nilpotent groups and later developments from them ( [12], [2], [3], [9], [10]). Schmidt proved that if all the maximal subgroups of a group G are nilpotent, then G is soluble, and that, in addition, if G is not nilpotent, |G| has exactly two distinct prime factors, G has a normal Sylow subgroup and a cyclic non-normal Sylow subgroup.…”

On the abnormal structure of finite groups

“…Rose [16] considered the effects of replacing "maximal" by "non-normal maximal" in Schmidt's result, and proved:…”

Maximal nonnormal chains in finite groups

“…Note that a theorem due to Rose [5] shows that h(G) = 2 implies solvability for G. More generally, we can effectively duplicate the proofs of the theorems in [2] to prove: Note that A 6 , the simple group of order sixty, has h(A δ ) = 4. The groups described in Theorem 5 have the property that they can be generated by two elements.…”