Frattini子群循环的有限$p$-群中的非交换集和极大Abel子群

设G是一个群,X是G的一个子集,若对于任意x,y∈X且x≠y,都有xy≠yx,则称X是G的一个非交换集.进一步,如果对于G中的任意其它非交换子集Y,都有|X|≥|Y|,那么称X是G的一个极大非交换集.文中确定了Frattini子群循环的有限p-群中极大非交换集和极大Abel子群的势.     

Weierstrass points on X 0(p ℓ) and arithmetic properties of Fourier coefficients of cusp forms

Generally it is unknown, whether or not ∞ is a Weierstrass point on the modular curve X ₀(N) if N is squarefree. A classical result of Atkin and Ogg states that ∞ is not a Weierstrass point on X ₀(N), if N=pM with p prime, p ∤ M and the genus of X ₀(M) zero. We use results of Kohnen and Weissauer to show that there is a connection between this question and the p-adic valuation of cusp forms under the Atkin–Lehner involution. This gives, in a sense, a generalization of Ogg’s Theorem in some cases.      

Principe de densité pour les groupes rèuctifs

The main goal of this Letter is to give a new proof of a result conjectured by D. De George and N. Wallach and proved by P. Deborne and to extend it to the S-arithmetic setting. The proof is based on a density principle.     

与正规性相关的各种子群之间的关系

本讨论了正规子群，拟正规子群，s-拟正规子群，半正规子群，c-正规子群，次正规子群，共轭置换子群，半置换子群，s-半置换子群这九种子群之间的关系。     

Cuccia定理的一个推广

For a finite group G,let S(G)be the set of minimal subgroups of odd order of G which are complemented in G.It is proved that if every minimal subgroup X of odd order of G which does not belong to S(G),C_G(X)is either subnormal or abnormal in G.Then G solvable.     

高维Mbius群的分类、生成系统和离散准则

本文利用高维Ｍｏｂｉｕｓ变换的Ｃｌｉｆｆｏｒｄ矩阵表示，给出了高维Ｍｏｂｉｕｓ子群的一个分类定理，证明了非初等子群的最小纯斜驶生成系统的存在性，得到了一条离散准则。     

Lower Bounds on the Number of Cyclic Subgroups in Finite Non-Cyclic Nilpotent Groups

Let $G$ be a finite group and $\mathfrak{c}(G)$ denote the number of cyclic subgroups of $G$. It is known that the minimal value of $\mathfrak{c}$ on the set of groups of order $n$, where $n$ is a positive integer, will occur at the cyclic group $Z_n$. In this paper, for non-cyclic nilpotent groups $G$ of order $n$, the lower bounds of $\mathfrak{c}(G)$ are established.     

Finite <Emphasis Type="Italic">p</Emphasis>-groups with a class of complemented normal subgroups

Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in Φ(G) have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified.     

Local commensurability graphs of solvable groups

The commensurability index between two subgroups A,B of a group G is [A:A∩B][B:A∩B]. This gives a notion of distance among finite index subgroups of G, which is encoded in the p-local commensurability graphs of G. We show that for any metabelian group, any component of the p-local commensurabilty graph of G has diameter bounded above by 4. However, no universal upper bound on diameters of components exists for the class of finite solvable groups. In the appendix we give a complete classification of components for upper triangular matrix groups in GL(2,𝔽_q).