Finite Groups Whose Minimal Subgroups are Weakly -SUBGROUPS (original) (raw)

Finite groups with ℋ-subgroups or strongly closed subgroups

Journal of Group Theory, 2012

Let G be a finite group. Goldschmidt, Flores and Foote investigated the following concept: A subgroup A of S is said to be strongly closed in S with respect to G if for every a A A, every element of S that is fused in G to a lies in a G. In particular, when A is of primepower order and S is a Sylow subgroup containing it, A is simply said to be a strongly closed subgroup. Bianchi, Mauri, Herzog and Verardi called a subgroup H of G an H-subgroup if N G ðHÞ V H g c H for all g A G. In fact, an H-subgroup of prime power order is the same as a strongly closed subgroup. In this paper, groups with certain H-subgroups of prime order are studied. For example, we give new characterizations of finite minimal non-T-groups (a T-group is a group for which normality is a transitive relation) by H-subgroups.

On weakly closed subgroups of finite groups

Monatshefte für Mathematik, 2014

Suppose that G is a finite group and H , K are subgroups of G. We say that H is weakly closed in K with respect to G if, for any g ∈ G such that H g ≤ K , we have H g = H. In particular, when H is a subgroup of prime-power order and K is a Sylow subgroup containing it, H is simply said to be a weakly closed subgroup of G or weakly closed in G. In the paper, we investigate the structure of finite groups by means of weakly closed subgroups.

A Note on Finite Groups Whose all of Subgroups are C-Normal

Proceedings of the 2013 International Conference on Advanced Computer Science and Electronics Information, 2013

A subgroup H of a group G is said to be-normal in G if there exists a normal subgroup N of G such that HN=G and H∩N ≤ Core(H) where Core(H) is the largest normal subgroup of G contained in H. In this paper we consider finite p-groups of order at most p 4 where p is a prime and show that all of their subgroups are c-normal. Also we study some classes of finite groups whose all of subgroups are c-normal.

Finite Groups with Minimal CSS-subgroups

European Journal of Pure and Applied Mathematics, 2021

Let G be a finite group. A subgroup H of G is called SS-quasinormal in G if there is a supplement B of H to G such that H permutes with every Sylow subgroup of B. A subgroup H of G is called CSS-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩K is SS-quasinormal in G. In this paper, we investigate the influence of minimal CSS-subgroups of G on its structure. Our results improve and generalize several recent results in the literature.

Subgroups of Finite Groups with a Strong Cover-Avoidance Property

Bulletin of the Australian Mathematical Society, 2009

A subgroup A of a group G has the strong cover-avoidance property in G, or A is a strong CAP-subgroup of G, if A either covers or avoids every chief factor of every subgroup of G containing A. The main aim of the present paper is to analyse the impact of the strong cover and avoidance property of the members of some relevant families of subgroups on the structure of a group.

On minimal non-p-closed groups and related properties

Publicationes Mathematicae Debrecen, 2011

Let p be a prime. A group is called p-closed if it has a normal Sylow p-subgroup and it is called p-exponent closed if the elements of order dividing p form a subgroup. A group is minimal non-p-closed if it is not p-closed but its proper subgroups and homomorphic images are. Similarly, a group is called minimal non-p-exponent closed if it is not p-exponent closed but all its proper subgroups and homomorphic images are. In this paper we characterize finite minimal non-p-closed groups and investigate the relationship between them and minimal non-p-exponent closed groups. In particular, we show that every minimal non-p-closed group is non-p-exponent closed and that minimal non-p-closed groups and simple minimal non-p-exponent closed groups have cyclic Sylow p-subgroups. Furthermore, given a prime p, we describe non-p-exponent closed groups of smallest order and we show that they coincide with non-p-closed groups of smallest order.

Subgroups close to normal subgroups

Journal of Algebra, 1989

Let G be a group and H a subgroup. It is shown that the set of indices {[H:HngHg~l] | geG) has a finite upper bound n if and only if there is a normal subgroup N < G which is commensurable with //; i.e., such that [H:Nr\H] and [N:Nr>H] are finite; moreover, the latter indices admit bounds depending only on n. If the bounded index hypothesis is assumed only for g running over some subgroup K<G, the conclusion holds with "normal" weakened to "normalized by K". More detailed Information is gölten under the assumption that {[//://ngHg~l] | geG} = {!,/>} for p prime. In particular, when p = 2 there exists N < G such that either H has index 2 in 7V, or N has index 2 in H. 1. THE {l,p} CASE This section contains the results referred to in the second paragraph of the abstract. The main results of the first paragraph are obtained in §2, which may be read independently. In the last three sections we extend both sets of results to the "normalized by K" context, obtain some modified bounds, and note some examples. We begin with the result that started this investigation. THEOREM l. Let G be a group, and H a subgroup. Then [H:Hr\gHg~l] <2 for all geG if and only if G has a normal subgroup N such that either (a) H < N and [N:H] < 2, or (b) N < H and [H:N] <2. Proof. "If" is clear. We shall prove "only if". Let X denote the set of conjugates of H. We observe that there are no proper inclusion relations among members of X, since if χ properly contained gxg~l, the latter would have index 2 in x, hence g 2 xg~ would have index 4 in x, a contradiction. Thus, if two members of X are distinct, their intersection has index 2 in each. Let us now choose any xeX and define two equivalence relations, [x] and , on X-{x}, letting y[x}z mean yr\x = zr>x, and yz mean yx = zx (equality of subsets). We Claim *This work was done while the authors were partly supported by NSF contracts DMS 85-02330 and DMS 87-06176 respectively. Electronic mail addresses of the authors at cartan.Berkeley.Edu or ucbcarta.bitnet: Bergman: gbergman, Lenstra: hwl.