Factorizations of one-generated -local formations.
Finite Groups Generated by 3-Transpositions. I.
Finite groups in which every self-centralizing subgroup is nilpotent or subnormal or a TI-subgroup
Let be a finite group. We prove that if every self-centralizing subgroup of is nilpotent or subnormal or a TI-subgroup, then every subgroup of is nilpotent or subnormal. Moreover, has either a normal Sylow -subgroup or a normal -complement for each prime divisor of .
Finite groups in which every two elements generate a soluble subgroup.
Finite groups in which subnormalizers are subgroups
Finite groups in which Sylow normalizers have nilpotent Hall supplements.
Finite groups in which -quasinormality is a transitive relation
Finite groups whose all proper subgroups are -groups
A group is said to be a -group if for every divisor of the order of , there exists a subgroup of of order such that is normal or abnormal in . We give a complete classification of those groups which are not -groups but all of whose proper subgroups are -groups.
Finite groups whose set of numbers of subgroups of possible order has exactly 2 elements
Counting subgroups of finite groups is one of the most important topics in finite group theory. We classify the finite non-nilpotent groups whose set of numbers of subgroups of possible orders has exactly two elements. We show that if is a non-nilpotent group whose set of numbers of subgroups of possible orders has exactly 2 elements, then has a normal Sylow subgroup of prime order and is solvable. Moreover, as an application we give a detailed description of non-nilpotent groups with...
Finite groups with -quasinormal subgroups.
Finite groups with few conjugacy classes of subgroups
Finite groups with Hall supplements to primitive subgroups.
Finite groups with primitive Sylow normalizers
We prove that are primitive the finite groups whose normalizers of the Sylow subgroups are primitive. We classify the groups of such class, denoted by , and we study the Schunck classes whose boundary is contained in . We give also necessary and sufficient conditions in order that the projectors be subnormally embedded.
Finite Groups with Small Conjugacy Classes.
Finite Groups with some -Permutably Embedded and Weakly -Permutable Subgroups
Let be a finite group, the smallest prime dividing the order of and a Sylow -subgroup of with the smallest generator number . There is a set of maximal subgroups of such that . In the present paper, we investigate the structure of a finite group under the assumption that every member of is either -permutably embedded or weakly -permutable in to give criteria for a group to be -supersolvable or -nilpotent.
Finite groups with some SS-supplemented subgroups
A subgroup of a finite group is said to be SS-supplemented in if there exists a subgroup of such that and is S-quasinormal in . We analyze how certain properties of SS-supplemented subgroups influence the structure of finite groups. Our results improve and generalize several recent results.
Finite groups with -subnormal conditions.
Finite groups with subnormal Shmidt subgroups.
Finite groups with weakly -semipermutable subgroups
Finite Groups with Weakly s-Permutably Embedded and Weakly s-Supplemented Subgroups
Suppose G is a finite group and H is a subgroup of G. H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup of G contained in H such that G = HT and ; H is called weakly s-supplemented in G if there is a subgroup T of G such that G = HT and , where is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We investigate the influence of the existence of s-permutably embedded and weakly s-supplemented...