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Abstract
A subgroup H of G is c-permutable in G if there exists a permutable subgroup P of G such that HP=G and H∩P≦H pG , where H pG is the largest permutable subgroup of G contained in H. A group G is called CPT-group if c-permutability is transitive in G. A number of new characterizations of finite solvable CPT-groups are given.
Abstract
Two sufficient conditions for a finite group G to be p-supersolvable have been obtained. For example (Theorem 1.1), let N be a normal subgroup of G such that G/N is p-supersolvable for a fixed odd prime p and let N p be a Sylow p-subgroup of N. Suppose that N is p-solvable and Ω1(N p) is generated by the subgroups of order p of N p which are normal in N G(N p). Then G is p-supersolvable.
Abstract
Let G be a finite group. A PT-group is a group G whose subnormal subgroups are all permutable in G. A PST-group is a group G whose subnormal subgroups are all S-permutable in G. We say that G is a PTo-group (respectively, a PSTo-group) if its Frattini quotient group G/Φ(G) is a PT-group (respectively, a PST-group). In this paper, we determine the structure of minimal non-PTo-groups and minimal non-PSTo-groups.
Abstract
This paper gives a characterization of finite groups G in which each cyclic subgroup either is normal in G or normalizes all subgroups of G.
Abstract
This paper represents an attempt to extend and improve the following result of Berkovich: Let G be a group of odd order. Let G=G 1 G 2 such that G 1 and G 2 are subgroups of G. If the Sylow p-subgroups of G 1 and of G 2 are cyclic, then G is p-supersolvable.
A subgroup H of G is called M p -embedded in G, if there exists a p-nilpotent subgroup B of G such that H p ∈ Syl p (B) and B is M p -supplemented in G. In this paper, we use M p -embedded subgroups to study the structure of finite groups.
Abstract
A number of authors have studied the structure of a finite group G under the assumption that some subgroups of G are well located in G. We will generalize the notion of s-permutable and s-permutably embedded subgroups and we will obtain new criterions of p-nilpotency and supersolvability of groups. We also generalize some known results.
Abstract
Let 
be a class of groups and G a finite group. We call a set Σ of subgroups of G a G-covering subgroup system for
if 
whenever 
. Let p be any prime dividing |G| and P a Sylow p-subgroup of G. Then we write Σ
p
to denote the set of subgroups of G which contains at least one supplement to G of each maximal subgroup of P. We prove that the sets Σ
p
and Σ
p
∪Σ
q
, where q≠p, are G-covering subgroup systems for many classes of finite groups.
