On fundamental groups of compact Hausdorff spaces

We discuss which groups can be realized as the fundamental groups of compact Hausdorff spaces. In particular, we prove that the claim ``every group can be realized as the fundamental group of a compact Hausdorff space' is consistent with the Zermelo-Fraenkel-Choice set theory.

     

On permuteral subgroups in finite groups

The permutizer of a subgroup H in a group G is defined as the subgroup generated by all cyclic subgroups of G that permute with H. Call H permuteral in G if the permutizer of H in G coincides with G; H is called strongly permuteral in G if the permutizer of H in U coincides with U for every subgroup U of G containing H. We study the finite groups with given systems of permuteral and strongly permuteral subgroups and find some new characterizations of w-supersoluble and supersoluble groups.     

On characterized subgroups of compact abelian groups

Let X be a compact abelian group. A subgroup H of X   is called characterized if there exists a sequence u=(_un)$\mathbf{u}=(u_n)$ of characters of X   such that H=_su(X)$H=s_{\mathbf{u}}(X)$, where _su(X):={x∈X:(_un,x)→0 in T}$s_{\mathbf{u}}(X):=\{x\in X:(u_n,x)\to 0\text{in}\mathbb{T}\}$. Every characterized subgroup is an _Fσδ$F_{\sigma \delta }$-subgroup of X  . We show that every _Gδ$G_{\delta }$-subgroup of X is characterized. On the other hand, X   has non-characterized _Fσ$F_{\sigma }$-subgroups.     

On sse-embedded subgroups of finite groups

Abstract

In this paper, we introduce the concept of sse-embedded subgroups of finite groups and present some new characterizations of solubility of finite groups using the sse-embedding property of subgroups. Furthermore, we discuss the sse-embedded subgroups in finite nonabelian simple groups. Some previously known results are generalized and unified.     

On g-s-supplemented subgroups of finite groups

A subgroup H of a group G is said to be g-s-supplemented in G if there exists a subgroup K of G such that HK⊴G and H ∩ K ⩽ H _sG , where H_sG is the largest s-permutable subgroup of G contained in H. By using this new concept, we establish some new criteria for a group G to be soluble.     

有限群的ss-置换子群

设G为有限群,称G的子群H为ss-置换子群,如果存在G的次正规子群B使得G=HB,且H与B的任意Sylow子群可以交换,即对任意X∈Syl（B）有XH=HX.利用子群的ss-置换性来研究有限群的结构,得到有限群超可解的两个充分条件.     

On permutable subgroups of finite groups

Let \frak Z \frak Z be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, \frak Z \frak Z contains exactly one and only one Sylow p-subgroup of G. A subgroup H of a finite group G is said to be \frak Z \frak Z -permutable if H permutes with every member of \frak Z \frak Z . The purpose here is to study the influence of \frak Z \frak Z -permutability of some subgroups on the structure of finite groups. Some recent results are generalized.     

On minimal subgroups of finite groups

In this paper, the minimal subgroups of a finite group G are studied. Using the normalizes of Sylow subgroups of G, we obtain some sufficient conditions for G to be p-milpotent and supersolvable, which ae extensions of the related results of Burnside, Ito, Buckley and Asaad.     

On s-semipermutable subgroups of finite groups

Suppose that G is a finite group and H is a subgroup of G. We say that H is ssemipermutable in G if HGv = GpH for any Sylow p-subgroup Gp of G with （p, ｜H｜） = 1. We investigate the influence of s-semipermutable subgroups on the structure of finite groups. Some recent results are generalized and unified.     

On complemented subgroups of finite groups

In this paper, it is proved that the class of all finite supersoluble groups with elementary abelian Sylow subgroups is just the class of all finite groups for which every minimal subgroup is complemented. The structure of a finite group under the assumption that all maximal subgroups (respectively 2-maximal) of any Sylow subgroup are complemented is also analyzed.     

On complemented subgroups of finite groups

A subgroup H of a group G is said to be complemented in G if there exists a subgroup K of G such that G = HK and H ⋂ K = 1. In this paper we determine the structure of finite groups with some complemented primary subgroups, and obtain some new results about p-nilpotent groups.