On the Marginal Automorphisms of a Group

Let W be a nonempty subset of a free group. We call an automorphism α of a group G a marginal automorphism if x ^?1α(x) ∈ W*(G) for each x ∈ G, where W*(G) is the marginal subgroup of G. In this article, we give some results on marginal automorphisms of a group.     

Nearly s-Normality of Groups and Its Properties

We call a subgroup H of a group G nearly s-normal in G if there exists N ? G such that HN ? G and H ∩ N ≤ H _sG , where H _sG is the largest s-permutable subgroup of G contained in H. In this article, we obtain some results about the nearly s-normal subgroups and use them to characterize the structure of finite groups.     

The Average Sylow Multiplicity Character and the Solvable Residual

Let G be a finite group, and let p₁,…, p_m be the distinct prime divisors of |G|. Given a sequence 𝒫 =P₁,…, P_m, of Sylow p_i-subgroups of G, and g ∈ G, denote by m_𝒫(g) the number of factorizations g = g₁…g_m such that g_i ∈ P_i. The properly normalized average of m_𝒫 over all 𝒫 is a complex character over G whose kernel contains the solvable radical of G [7 Levy , D. ( 2010 ). The average Sylow multiplicity character and solvability of finite groups . Communications in Algebra. 38 : 632 – 644 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. The present paper characterizes the solvable residual of G in terms of this character.     

Finite p-Groups All of Whose Minimal Nonnormal Subgroups Posses Large Normal Closures

We obtained some results about finite p-groups G with G/H^G being abelian for all nonnormal subgroups H, where H^G denotes the normal closure of H. Moreover, we give a classification of finite p-groups G with G/H^G being cyclic for all nonnormal subgroups H.     

On Abelian Groups Having All Proper Fully Invariant Subgroups Isomorphic

We introduce two classes of abelian groups which have either only trivial fully invariant subgroups or all their nontrivial (respectively nonzero) fully invariant subgroups are isomorphic, called IFI-groups and strongly IFI-groups, such that every strongly IFI-group is an IFI-group, respectively. Moreover, these classes coincide when the groups are torsion-free, but are different when the groups are torsion as well as, surprisingly, mixed groups cannot be IFI-groups. We also study their important properties as our results somewhat contrast with those from [13 Grinshpon , S. Ya. , Nikolskaya (Savinkova) , M. M. ( 2011 ). Fully invariant subgroups of abelian p-groups with finite Ulm-Kaplansky invariants . Commun. Algebra 39 : 4273 – 4282 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] and [14 Grinshpon , S. Ya. , Nikolskaya (Savinkova) , M. M. ( 2011-2012/2014 ). Torsion IF-groups . Fundam. Prikl. Mat. 17 : 47 – 58 ; translated in J. Math. Sci. 197:614–622 . [Google Scholar]].     

Sylow Products and the Solvable Residual

We study the connection between products of Sylow subgroups of a finite group G and the solvable residual of G. Let Π(𝒫) be a product of Sylow subgroups of G such that each prime divisor of |G| is represented exactly once in Π(𝒫). We prove that there exists a unique normal subgroup N of G which is minimal subject to the requirement Π(𝒫) N = G. Furthermore, N is perfect, and the product of all of these subgroups is the solvable residual of G. We also prove that the solvable residual of G is generated by all elements which arise from non-trivial factorizations of 1 _G in such products of Sylow subgroups.     

非线性不可约特征标维数相等且为素数个的有限群

对于具有素数个非线性不可约特征标且它们的维数相等的有限群,我们给出一个分类.     

具有指数为2的循环子群的亚循环群

我们给出了具有指数为2的循环子群的亚循环群的同构分类.这个结果可应用于正则地图的研究中.     

关于有限群的$CAP$-嵌入子群

A subgroup H of a finite group G is said to be CAP-embedded subgroup of G if, for each prime p dividing the order of H, there exists a CAP-subgroup K of G such that a Sylow p-subgroup of H is also a Sylow p-subgroup of K. In this paper some new results are obtained based on the assumption that some subgroups of prime power order have the CAP-embedded property in the group.