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A subgroup A of a finite group G is called a TI-subgroup if either A ∩ A x = 1 or A ∩ A x = A holds for all x ∈ G. In this paper, finite group all of whose meta-cyclic subgroups are TI-subgroups are classified completely. In particular, such groups are solvable.

In this paper, we characterize finite groups G satisfying that, for every prime power divisor p n of the order of G, there exists a subgroup H of index p n (of order p n ) in G such that H is normal or abnormal in G.

A subgroup H of a group G is called a CAP*-subgroup of G, if H either covers or avoids every non-Frattini chief factor of G. In this article, we give some interesting properties of CAP*-subgroups. Furthermore, we determine the structure of finite groups based on the assumption that some subgroups are CAP*-subgroups and generalize some known results.

A subgroup H of a finite group G is called Hall normally embedded in G if H is a Hall subgroup of the normal closure H G . Groups which contain a Hall normally embedded subgroup of order d for every factor d of | G | are characterized. Such groups are supersolvable with a cyclic nilpotent residual of square-free order.

A group G is called an “ℋ-QC-group” if for any element x of order 2 or 4 of G, ⟨x⟩ ⟨y⟩ = ⟨y⟩ ⟨x⟩ for all y in G. In this article, we investigate the structure of the groups G of the title. Suppose that G is non-2-closed and non-2-nilpotent, then it turns out that such a group is either S 4 or a uniquely determined group of order 48 or A 5 or SL(2,5). The corresponding problem for G 2-closed or 2-nilpotent is open but very difficult.

A subgroup of a group G is said to be Sylow-quasinormal (S-quasinormal) in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be Supplement-Sylow-quasinormal (SS-quasinormal) in G if there is a supplement B of H to G such that H is permutable with every Sylow subgroup of B. In this article, we investigate the influence of SS-quasinormal of maximal or minimal subgroups of Sylow subgroups of the generalized Fitting subgroup of a finite group.

Let d be the smallest generator number of a finite p-group P and let ℳ d (P) = {P 1,…, P d } be a set of maximal subgroups of P such that . In this article, the structure of a finite group G under some assumptions on the S-quasinormal and normal embedding of subgroups in ℳ d p (G p ), for each prime p, and Sylow p-subgroups G p of G is investigated.