Outer-\(\Sigma\) groups of finite order (Q1105693)

Suppose \(\Sigma\) is a group-theoretic property. A group with property \(\Sigma\) is called a \(\Sigma\) group. A group which is not a \(\Sigma\) group is called an inner-\(\Sigma\) group if every proper subgroup of it is a \(\Sigma\) group and an outer-\(\Sigma\) group if every proper factor group of it is a \(\Sigma\) group. The Principal Lemma in the paper deals with groups having trivial Frattini subgroup and a unique minimal normal subgroup which is elementary abelian. It is then proved that certain classes of groups possess the property stated in this Principal Lemma; one such example is any solvable outer-\(\Sigma\) group when the class of all \(\Sigma\) groups is a saturated formation. By using this Principal Lemma and some consequences the nilpotent groups of class less than a certain number k, called c(k)-groups, the groups for which the k-th term of the lower central series is p-nilpotent, called \(\Gamma_ k\)-pn groups, and p-supersolvable groups are discussed.