A way of extending theorems in group theory (Q687948)

Using his previous results on inner and outer \(\Sigma\)-groups and on minimal non-\(\Sigma\)-groups [J. Southwest Teach. Univ. (1988)], the author proves two general theorems which enable him to generalize results of Itô, Huppert, Kramer, Baer and the reviewer. As a sample, we quote the following generalization of one of Kramer's results: Let \(G\) be a finite solvable group and let \(F_ n(G)\) be the \(n\)th term of its Fitting series. If \(| F_ k(G): L \cap F_ k(G)|=1\) or a prime for some \(k \geq 1\) and every maximal subgroup \(L\) of \(G\), then \(G\) is supersolvable (Kramer's Theorem is just the case \(k = 1\)).