Structure of finite groups with S-quasinormal third maximal subgroups

Abstract

All groups considered in the present paper are finite. Recall that a subgroup H of a group G is called a 2-maximal subgroup (or a second maximal subgroup) of the group G if H is a maximal subgroup in a certain maximal subgroup M of the group G: By analogy, one can define 3-maximal subgroups, 4-maximal subgroups, etc. It is easy to see that, in nonsupersolvable groups, a subgroup can be both n-maximal and m-maximal simultaneously for n ¤ m: In this connection, we say that a subgroup H of a group G is a strictly n-maximal subgroup in G if H is an n-maximal subgroup in G but is not an n-maximal subgroup in any proper subgroup of G: For example, in the group SL.2; 3/; the unique subgroup of order 2 is 2-maximal but not strictly 2-maximal. The relationship between n-maximal semigroups .n > 1/ of a group G and the structure of the group G was studied by many authors. Apparently, the first result in this direction was obtained by Huppert [1], who proved that a group is supersolvable if all 2-maximal subgroups of it are normal. In the same paper, it was also proved that if each third maximal subgroup of a group G is normal in G; then the commutant G of the group G is nilpotent, and the order of each principal factor of the group G is divided by at most two (not necessarily different) prime numbers. Huppert’s work [1] stimulated numerous investigations in this direction. In particular, developing Huppert’s results, Janko obtained a description of groups in which 4-maximal subgroups are normal. He proved that if each 4-maximal subgroup of a solvable group G is normal in G and the order of G is divided by at most four different prime numbers, then G is a supersolvable group [2]. A year later, Janko studied groups that do not contain 5maximal subgroups other than the identity one [3]. Among the early works in this direction, we also note Agrawal’s work [4], where it was proved that a group is supersolvable if each 2-maximal subgroup of it is S -quasinormal (a subgroup H of a group G is called S -quasinormal, or S -permutable, in G if H is permutable with all Sylow subgroups of G). The results of Huppert and Janko were also naturally developed by Mann [5], who analyzed the structure of groups in which each n-maximal subgroup is subnormal. Later, Asaad [6] improved the results of Huppert and Janko for strictly n-maximal subgroups with n D 2; 3; 4: In the present paper, we give a complete description of groups all 3-maximal subgroups of which are S quasinormal. Based on this result, we also solve, in the class of nonnilpotent groups, the Huppert problem of complete description of groups in which 3-maximal subgroups are normal.