Subgroups Of Finite Groups Research Articles

Determining Hypercentral Hall Subgroups in Finite Groups

Let G be a finite group, and let π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document} be a set of primes. The aim of this paper is to obtain some results concerning how much information about the π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document}-structure of G can be gathered from the knowledge of the sizes of conjugacy classes of its π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document}-elements and of their multiplicities. Among other results, we prove that this multiset of class sizes determines whether G has a hypercentral Hall π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document}-subgroup.

Union of products of subgroups in finite groups

Let G be a group and A, B, C be subgroups of G. In this paper, we prove that if G = AB ∪ BC , then G = AB or G = BC. But, in general, G = AB ∪ AC does not imply that G = AB or G = AC. We also prove that if G is a 2-decomposed group, and G is a Sylow tower group or G is nilpotent-by-nilpotent, then G = AB ∪ AC implies that G = AB or G = AC. Other interesting special cases are also discussed.

On Intersections of Nilpotent Subgroups in Finite Groups with Simple Socle from the “Atlas of Finite Groups”

On Intersections of Nilpotent Subgroups in Finite Groups with Simple Socle from the “Atlas of Finite Groups”

σ-Subnormality in locally finite groups

Let σ={σj:j∈J} be a partition of the set P of all prime numbers. A subgroup X of a finite group G is σ-subnormal in G if there exists a chain of subgroupsX=X0≤X1≤…≤Xn=G such that, for each 1≤i≤n−1, Xi−1⊴Xi or Xi/(Xi−1)Xi is a σji-group for some ji∈J. Skiba [18] studied the main properties of σ-subnormal subgroups in finite groups and showed that the set of all σ-subnormal subgroups plays a very relevant role in the structure of a finite soluble group. In this paper we lay the foundation of a general theory of σ-subnormal subgroups (and σ-series) in locally finite groups. Although in finite groups, σ-subnormal subgroups form a sublattice of the lattice of all subgroups (see for instance [3]), this is no longer true for locally finite groups; in fact, the join of σ-subnormal subgroups is not always σ-subnormal, but this is the case (for example) whenever the join of subnormal subgroups is subnormal (see Theorem 3.16). We provide many criteria to determining when a subgroup is σ-subnormal starting from the much weaker concept of σ-seriality (see Section 2). These criteria are particularly useful when employed to investigate the join of σ-subnormal subgroups — we show for example that if two σ-subnormal subgroups H and K of a locally finite group G are such that HK=KH, then HK is σ-subnormal in G (see Theorem 3.15) — but they are also fit to show that on some occasions σ-seriality coincides with σ-subnormality — this is the case of linear groups (see Theorem 3.35).

On Subpermuteral Subgroups in Finite Groups

On Subpermuteral Subgroups in Finite Groups

On Intersections of Certain Nilpotent Subgroups in Finite Groups

On Intersections of Certain Nilpotent Subgroups in Finite Groups

Products of F∗(G)-subnormal subgroups of finite groups

A subgroup [Formula: see text] of a group [Formula: see text] is called [Formula: see text]-subnormal if it is subnormal in the product with the generalized Fitting subgroup [Formula: see text] of [Formula: see text]. All saturated formations [Formula: see text] that contain every group [Formula: see text] where [Formula: see text] and [Formula: see text] are [Formula: see text]-subnormal [Formula: see text]-subgroups are described. Moreover, if such formation [Formula: see text] is also normally hereditary, then every group [Formula: see text] where [Formula: see text] and [Formula: see text] are [Formula: see text]-subnormal quasi-[Formula: see text]-subgroups is a quasi-[Formula: see text]-group. The connection of above-mentioned formations to the lattice formations, GWP-formations and formations with the Kegel property is found.

On strongly closed and Hall s-semiembedded subgroups of finite groups

Let [Formula: see text] be subgroups of a finite group [Formula: see text], then [Formula: see text] is called strongly closed in [Formula: see text] with respect to [Formula: see text] if [Formula: see text] for every [Formula: see text], and in particular, [Formula: see text] is simply called strongly closed in [Formula: see text] if [Formula: see text] is strongly closed in [Formula: see text] with respect to [Formula: see text]. Let [Formula: see text] be a subgroup of a finite group [Formula: see text], then [Formula: see text] is called Hall [Formula: see text]-semiembedded in [Formula: see text] if [Formula: see text] is a Hall subgroup of [Formula: see text] for every [Formula: see text], where [Formula: see text]. In this paper, we obtain some criteria for [Formula: see text]-nilpotency of a finite group and extend some known results concerning strongly closed and Hall [Formula: see text]-semiembedded subgroups. In particular, we generalize some main results of Guo and Li [Hall [Formula: see text]-semiembedded subgroups and [Formula: see text]-nilpotency of finite groups, Southeast Asian Bull. Math. 42(3) (2018) 367–374] and Kong [New characterizations of [Formula: see text]-nilpotency of finite groups, J. Algebra Appl. 20(11) (2021) Article ID: 2150215, 6 pp.].

A note on S-semipermutable and S-permutably embedded subgroups of finite groups

A note on S-semipermutable and S-permutably embedded subgroups of finite groups

On Π-Permutable Subgroups in Finite Groups

On Π-Permutable Subgroups in Finite Groups

Semipermutable subgroups and s-semipermutable subgroups in finite groups

Semipermutable subgroups and s-semipermutable subgroups in finite groups

On Intersections of Certain Nilpotent Subgroups in Finite Groups

Доказано, что в любой конечной группе $G$ с нильпотентными подгруппами $A$ и $B$ и условием $A\cap B^g\unlhd\langle A,B^g\rangle$ для любого $g$ из $G$ подгрупп $\operatorname{Min}_G(A,B)\le F(G)$. Это обобщает теорему автора о пересечениях абелевых подгрупп в конечной группе, так как справедливо, например, для гамильтоновых подгрупп $A$ и $B$ из $G$. Библиография: 7 названий.

On the Pronormality of Second Maximal Subgroups in Finite Groups with Socle $$L_{2}(q)$$

According to P. Hall, a subgroup \(H\) of a finite group \(G\) is called pronormal in \(G\) if, for any element \(g\) of \(G\), the subgroups \(H\) and \(H^{g}\) are conjugate in \(\langle H,H^{g}\rangle\). The simplest examples of pronormal subgroups of finite groups are normal subgroups, maximal subgroups, and Sylow subgroups. Pronormal subgroups of finite groups were studied by a number of authors. For example, Legovini (1981) studied finite groups in which every subgroup is subnormal or pronormal. Later, Li and Zhang (2013) described the structure of a finite group \(G\) in which, for a second maximal subgroup \(H\), its index in \(\langle H,H^{g}\rangle\) does not contain squares for any \(g\) from \(G\). A number of papers by Kondrat’ev, Maslova, Revin, and Vdovin (2012–2019) are devoted to studying the pronormality of subgroups in a finite simple nonabelian group and, in particular, the existence of a nonpronormal subgroup of odd index in a finite simple nonabelian group. In The Kourovka Notebook, the author formulated Question 19.109 on the equivalence in a finite simple nonabelian group of the condition of pronormality of its second maximal subgroups and the condition of Hallness of its maximal subgroups. Tyutyanov gave a counterexample \(L_{2}(2^{11})\) to this question. In the present paper, we provide necessary and sufficient conditions for the pronormality of second maximal subgroups in the group \(L_{2}(q)\). In addition, for \(q\leq 11\), we find the finite almost simple groups with socle \(L_{2}(q)\) in which all second maximal subgroups are pronormal.

On Fisher Subgroups of Finite Groups

On Fisher Subgroups of Finite Groups

On -permutable subgroups in finite groups

UDC 512.542 Let be some partition of the set of all primes and let be a nonempty subset of the set A set of subgroups of a finite group is said to be a \emph{complete Hall -set} of if every member of is a Hall -subgroup of for some and contains exactly one Hall -subgroup of for every such that A subgroup of is called (i) {-<em>permutable</em>} if for and ; (ii) {-<em>permutable</em> in } if is -permutable for some complete Hall -set of We study the influence of -permutable subgroups on the structure of In particular, we prove that if and where and are -permutable -separable (respectively, -closed) subgroups of then is also -separable (respectively, -closed). Some known results are generalized.

On generalised subnormal subgroups of finite groups

On generalised subnormal subgroups of finite groups

On p-nilpotence and $$IC{\Phi}$$-subgroups of finite groups

On p-nilpotence and $$IC{\Phi}$$-subgroups of finite groups

A question of zhou, shi and duan on nonpower subgroups of finite groups

A subgroup H of a group G is called a power subgroup of G if there exists a non-negative integer m such that H = ⟨gm : g ∈ G⟩. Any subgroup of G which is not a power subgroup is called a nonpower subgroup of G. Zhou, Shi and Duan, in a 2006 paper, asked whether for every integer k (k ≥ 3), there exist groups possessing exactly k nonpower subgroups. We answer this question in the affirmative by giving an explicit construction that leads to at least one group with exactly k nonpower subgroups, for all k ≥ 3, and in_nitely many such groups when k is composite and greater than 4. Moreover, we describe the number of nonpower subgroups for the cases of elementary abelian groups, dihedral groups, and 2-groups of maximal class.

On the generalized norms of finite groups

The norm [Formula: see text] of a group [Formula: see text] is the intersection of the normalizers of all subgroups in [Formula: see text]. In this paper, the norm is generalized by studying on Sylow subgroups and [Formula: see text]-subgroups in finite groups which is denoted by [Formula: see text] and [Formula: see text], respectively. It is proved that the generalized norms [Formula: see text] and [Formula: see text] are all equal to the hypercenter of [Formula: see text].

Some notes on the second maximal subgroups of finite groups

Some notes on the second maximal subgroups of finite groups