On Subgroups Generated by Small Classes in Finite Groups

Let G be a finite group and M(G) be the subgroup of G generated by all noncentral elements of G that lie in the conjugacy classes of the smallest size. Recently several results have been proved regarding the nilpotency class of M(G) and F(M(G)), where F(M(G)) denotes the Fitting subgroup of M(G). We prove some conditional results regarding the nilpotency class of M(G).     

Finite Groups with Conjugacy Classes Number One Greater than Its Same Order Classes Number

ABSTRACT

Let k(G) be the number of conjugacy classes of finite groups G and π _e (G) be the set of the orders of elements in G. Then there exists a non-negative integer k such that k(G) = |π _e (G)| + k. We call such groups to be co(k) groups. This article classifies all finite co(1) groups. They are isomorphic to one of the following groups: A ₅, L ₂(7), S ₅, Z ₃, Z ₄, S ₄, A ₄, D ₁₀, Hol(Z ₅), or Z ₃ ? Z ₄.     

Finite Groups with Few Non-Abelian Subgroups

Let G be a finite group and τ(G) denote the number of conjugacy classes of all non-abelian subgroups of G. The symbol π(G) denotes the set of the prime divisors of |G|. In this paper, finite groups with τ(G) ≤ |π(G)| are classified completely. Furthermore, finite nonsolvable groups with τ(G) = |π(G)| +1 are determined.     

一类和有限群的共轭类关联的新图

给出了满足下列性质P_4的有限群的分类,对任意4个不同的非中心的共轭类,它们的代表元的阶的公因子为1.     

The Influence of Some Quantitative Properties of Abelian Subgroups on Solvability of Finite Groups

As an important application of Thompson's theorem [9 Robinson , D. J. S. ( 1996 ). A Course in the Theory of Groups. , 2nd ed. New York : Springer-Verlag .[Crossref] , [Google Scholar], Theorem 10.4.2], a finite group is solvable if it has an abelian maximal subgroup. In this article, we mainly investigate the influence of some quantitative properties of abelian subgroups on solvability of finite groups. Some new results are obtained.     

Abelian Subgroups of Garside Groups

In this article, we show that for every abelian subgroup H of a Garside group, some conjugate g ^?1 Hg consists of ultra summit elements and the centralizer of H is a finite index subgroup of the normalizer of H. Combining with the results on translation numbers in Garside groups, we obtain an easy proof of the algebraic flat torus theorem for Garside groups and solve several algorithmic problems concerning abelian subgroups of Garside groups.     

Finite Groups with Given Quantitative Non-Nilpotent Subgroups II

As an extension of Shi and Zhang's 2011 article [4 Shi , J. , Zhang , C. ( 2011 ). Finite groups with given quantitative non-nilpotent subgroups . Comm. Algebra 39 : 3346 – 3355 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], we prove that any finite group having at most 23 non-normal non-nilpotent proper subgroups is solvable except for G ? A ₅ or SL(2, 5), and any finite group having at most three conjugacy classes of non-normal non-nilpotent proper subgroups is solvable except for G ? A ₅ or SL(2, 5).     

On Finite Groups in Which Cyclic Subgroups of the Same Order are Conjugate

We consider finite groups G for which any two cyclic subgroups of the same order are conjugate in G. We prove various structure results and, in particular, that any such group has at most one non-abelian composition factor, and this is isomorphic to PSL(2, p ^m ), with m odd if p is odd, or to Sz(2^2m+1), or to one of the sporadic groups M ₁₁, M ₂₃, or J ₁.     

非循环子群共轭类个数小于等于2的有限群

本文讨论了非循环子群共轭类个数小于等于2的有限群,给出了此类群的完全分类.     

On p-Regular G-Conjugacy Classes and the p-Structure of Normal Subgroups

Let N be a p-solvable normal subgroup of a group G such that N contains a noncentral Sylow r (≠ p)-subgroup R of G. It is proved that the p-complements of N are nilpotent if |x ^G |=1 or m for every p-regular element x of N whose order is divisible by at most two distinct primes. Our result, therefore, gives some information concerning the nilpotence of some kind of subgroups of a group G.     

On Finite Number of Conjugacy Classes in Groups

The work is inspired by an article of Herzog, Longobardi, and Maj, who considered groups with a finite number of infinite conjugacy classes. Their main results were obtained under assumption that the FC-center is of finite index in the group. We consider here infinite groups with a finite number of conjugacy classes of any size (FNCC-groups). Hence the FC-center in our case will be finite, but of infinite index in the group. Among results on these groups we give a criterion for a wreath product of FNCC-groups to be an FNCC-group.     

Conjugacy Classes Outside a Normal Subgroup

Abstract

In this paper, we investigate the structure of finite groups G admitting a normal subgroup N such that N contains almost all conjugacy classes of G.     

Finite Groups Having Exactly Two Conjugate Classes of Non-subnormal Subgroups

In this paper, finite groups having exactly two conjugate classes of non-subnormal subgroups of G are completely classified.     

Finite Solvable Groups with an Irreducible Character Vanishing on Just One Class of Elements

The aim of this note is to classify the finite solvable groups with an irreducible character that vanishes on just one class of elements.     

非幂零极大子群共轭类类数给定的有限群

证明了非幂零极大子群共轭类类数等于2的有限群必可解,并给出了非幂零极大子群同阶类类数等于2的非可解群的等价刻画.     

Classes of Subgroups of Simply Presented Primary Abelian Groups

A class 𝒳 of abelian p-groups is closed under ω₁-bijective homomorphisms if whenever f: G → H is a homomorphism with countable kernel and cokernel, then G ∈ 𝒳 iff H ∈ 𝒳. For an ordinal α, we consider the smallest class with this property containing (a) the p ^α-bounded simply presented groups; (b) the p ^α-projective groups; (c) the subgroups of p ^α-bounded simply presented groups. This builds upon classical results of Nunke from [14 Nunke , R. ( 1963 ). Purity and subfunctors of the identity . In: Topics in Abelian Groups , Chicago : Scott, Foresman and Co. , pp. 121 – 171 . [Google Scholar]] and [15 Nunke , R. ( 1967 ). Homology and direct sums of countable abelian groups . Math. Z. 101 : 182 – 212 .[Crossref], [Web of Science ®] , [Google Scholar]]. Particular attention is paid to the separable groups in these classes.     

On Lengths of Conjugacy Classes and Character Degrees in Finite Groups

We extend the two results of Dolfi and Isaacs to π-separable groups, where π = {p, q}, from solvable groups.     

恰有两个子群共轭类长的有限群

设G是有限群,Ns(G)表示G的子群共轭类长构成的集合.本文研究Ns(G)中只有两个元素时有限群G的结构,在非幂零情形时给出了G的完全分类,在幂零情形时获得了G的一些性质.