Sufficient conditions for the solvability and supersolvability in finite groups
A finite group G is called an H-r N-group if i) G has even order and ii) each even-ordered subgroup H of G with \(| H|\) the product of r not necessarily distinct primes, is normal in G. The author proves the following results: 1. If G is an H-2 N-group that does not involve \(A_ 4\), then G is supersolvable. 2. If G is an H-2 N-group or an H-3 N-group then G' is nilpotent. 3. If G is an H-4 N-group then either G is solvable or G is isomorphic with \(L_ 2(q)\) for certain odd prime powers q. The author also shows that if G is an odd order group in which each subgroup of order pq is normal (where p and q are not necessarily distinct primes) then G' is nilpotent. Finally, the author proves that a group G in which each subgroup of order \(p^ 2\) is normal, for every prime factor of \(| G|\) except perhaps the largest, is either Sylow-towered in the natural order or involves \(A_ 4\). The work is related to a paper by \textit{N. S. Sastry} and the reviewer on the influence of normality conditions on almost minimal subgroups [J. Algebra 52, 364-377 (1978; reviewed above)].
scientific article; zbMATH DE number 3865557
scientific article