Finite groups with non-nilpotent maximal subgroups

In this paper, the structure of a finite group under group theoretic restrictions on its non-nilpotent subgroups has been investigated.     

Finite groups with S-permutable n-minimal subgroups

We study the supersolvability of finite groups and the nilpotent length of finite solvable groups under the assumption that all their exactly n-minimal subgroups are S-permutable, where n is an arbitrary integer.     

Finite groups with seminormal Sylow subgroups

In this paper, we prove the following theorem： Let p be a prime number, P a Sylow psubgroup of a group G and π = π（G） ／ {p}. If P is seminormal in G, then the following statements hold： 1） G is a p-soluble group and P＇ ≤ Op（G）; 2） lp（G） ≤ 2 and lπ（G） ≤ 2; 3） if a π-Hall subgroup of G is q-supersoluble for some q ∈ π, then G is q-supersoluble.     

Finite groups with some pronormal subgroups

A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to H ^x in 〈H, H ^x 〉. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups (non-PRN-group all of whose proper subgroups are PRN-groups). At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups.     

Finite groups with some SS-supplemented subgroups

Czechoslovak Mathematical Journal - A subgroup H of a finite group G is said to be SS-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is S-quasinormal in K. We...     

Finite groups with two noncomplemented subgroups

Mathematical Notes - The finite groups containing exactly two noncomplemented subgroups are described.     

Finite groups with seminormal Schmidt subgroups

A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup A is said to be seminormal in a group G if there exists a subgroup B such that G = AB and AB₁ is a proper subgroup of G, for every proper subgroup B₁ of B. Groups that contain seminormal Schmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Schmidt {2, 3}-subgroups and all 5-closed {2, 5}-Schmidt subgroups of the group are seminormal; the classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary. Supported by BelFBR grant Nos. F05-341 and F06MS-017. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 448–458, July–August, 2007.