内交换的capable p-群性质研究

研究了内交换p-群G是capable群需要满足的条件,得到了这类群是capable群的充要条件.并由内交换p-群G构造得到了群H,使得H满足HH/Z(H)≌G.     

p~n(n≤4)阶capable群

一个群G能够充当另外一个群H的中商群,称群G是capable群.考虑的问题是:对于阶为p~n(n≤4)的p-群G,哪些群是capable群?完全决定了p~n(n≤4)阶的capable p-群G.     

有限群的p-幂零性和极小子群

从有限群构造的角度来看,p-幂零群和p-可分解群很相似,前者是p-群和p'-群的半直积,后者是p-群和P'-群的直积.有限群G的p-可分解剩余是P(G)=∩{H|H(⊿) G,而且G/H是p-可分解}.通过观察p-可分解剩余和极小子群的性质得到了几个关于p-幂零的充要条件.     

关于有限群的弱S-可补嵌入子群(英文)

假定H是有限群G的一个子群.如果对于|H|的每个素因子p,H的一个Sylow p-子群也是G的某个s-可换子群的Sylow p-子群,则称H为G的s-可换嵌入子群;如果存在G的子群T使得G=HT并且H∩T≤HG,其中HG为群G含于H的最大的正规子群,则称H为G的c-可补子群;如果存在G的子群T使得G=HT并且H∩T≤Hse,其中Hse为群G含于H的一个s-可换嵌入子群,则称H为G的弱s-可补嵌入子群.本文研究弱s-可补嵌入子群对有限群结构的影响.某些新的结论被进一步推广.     

有限群的弱S-拟正规嵌入子群

群G的子群H称为在G中S-拟正规嵌入的,如果对于任意的素数p||H|,H的Sylow p-子群也是G的某个S-拟正规子群的Sylow p-子群.称群G的子群H在G中弱S-拟正规嵌入,如果存在群G的正规子群T,使得HT■G且H∩T在G中是S-拟正规嵌入的.研究了弱S-拟正规嵌入子群的性质,给出了某些群类的新的特征,并推广了一些已知的结论.     

无限正则p-群

对一类无限正则p-群进行了研究,得到了一个正则的局部幂零p-群G如果满足|G(Ω)1(G)|＜∞,那么G是幂零的且G是可除阿贝尔p-群被有限群的扩张.进而,还研究了一类无限的非正则p-群,但它的所有真商群或者真的无限子群是正则群.在假设这类群存在拟循环子群的情况下,在定理1.2和1.3给出了这类群的结构的刻画.     

具有p-幂零s-补子群的有限群（英文）

有限群G的子群H叫做F-s-补子群,若存在G的一个子群K使得G=HK且K/(K∩H_G)∈F,其中F是一个群类.本论文利用p-幂零s-补子群得到了关于有限群为p-幂零群的一些新成果.     

p-Fitting子群的CAP-嵌入子群

群G的子群H称为G的CAP-嵌入子群,如果对于|H|的每个素因子p,存在G的某个CAP-子群K,使得H的某个Sylow p-子群也是K的一个Sylowp-子群.本文通过假定G的p-Fitting子群F_p(G)的某个Sylow p-子群的每个极大子群是G的CAP-嵌入子群,得到一些新的结果.     

S-拟正规嵌入子群

群G的一个子群H称为在G中S-拟正规嵌入的,如果对H阶中的每一个素因子p,H的Sylowp-子群也是G的某个S-拟正规子群的Sylowp-子群.利用子群的S-拟正规嵌入性给出了群为p-幂零群及超可解群的一些特征.     

p-超循环嵌入子群的一个判别准则

令E是有限群G的一个正规子群,且U是所有有限超可解群的集合.E称为在G中是p-超循环嵌入的,如果E的每个pd-阶的G-主因子是循环的.G的子群H称为在G中是U-Φ-可补充的,如果存在G的一个次正规子群T,使得G=HT,且(H∩T)H_G/H_G≤Φ/(H/H_G)Z_U(G/H_G),其中Z_U(G/H_G)是商群G/H_G的U-超中心.作者证明,如果E的一些p-子群在G中是U-Φ-可补充的,那么E在G中是p-超循环嵌入的.作为应用,得到了有限群是p-超可解的若干判断准则,并且推广了一些已知的结果.     

真商群为s-自对偶群的有限p-群

如果有限群G的每个子群与G的某个商群同构,则称群G为s-自对偶群.如果s-自对偶群G的每个商群与G的某个子群同构,则称群G为自对偶群.本文分类了每个真商群均为s-自对偶群的有限p-群.作为推论,本文还分类了每个真截段均为s-自对偶群的有限p-群,每个真商群均为自对偶群的有限p-群,以及每个真截段均为自对偶群的有限p-群.     

On the normal subgroup lattice of a hypercentral p-group

We describe all hypercentral p-groups G whose lattice of normal subgroups n(G) is isomorphic to n(H) for a group H with hypercenwal derived subgroup and H not a pgroup.     

On commutative group algebras of mixed groups

Suppose F is a perfect field of characteristic p 0 and G is an abelian group whose torsion subgroup Gt is p-primary. If Gt is totally projective of countable length, it is shown that G is a direct factor of the group of normalized units V(G) of the group algebra F(G) and that V(G)/G is a totally projective p-group. The proof of this result is based on a new characterization of the class of totally projective p-groups of countable length. Li addition, the same result regarding V(G) is obtained if G has countable torsion-free rank and Gt is totally projective of length less than ω₁ + ω₀ . Finally, these results are applied to the question of whether the existence of an F-i pomorphism F(G) ? F(H) for some group H implies that G?H.     

On Groups with a CC(n)-Subgroup

A subgroup H of G is a CC(n)-subgroup of G if |G: H| >n and |C_G(x): C_H(x)| ≤n for each element x ∈ H ? {1}. In this article, we study the finite p-groups with a nontrivial CC(p)-subgroup, and the locally nilpotent groups with a nontrivial CC(n)-subgroup.     

The Navarro–Tiep Galois conjecture for $$p=2$$ p = 2

We investigate the upper $$FC-$$ central series of the unit group of an integral group ring $${\mathbb Z}G$$ of a periodic group G. We prove that $${\mathcal U}={{\mathcal U}}_1({\mathbb Z}G)$$ has $$FC-$$ central height one if and only if the $$FC-$$ hypercenter of $${{\mathcal U}}_1({\mathbb Z}G)$$ is contained in the normalizer of the trivial units. Further, in these conditions, the $$FC-$$ hypercenter of the unit group is non-central if and only if G is a $$Q^{*}-$$ group. Let $$H \vartriangleleft {\mathcal U}, H$$ contained in the normalizer of the trivial units, suppose that either the elements of finite order form a subgroup or H is a polycyclic-by-finite (polycyclic) subgroup, then H is contained in the finite conjugacy center of $${{\mathcal U}}_1({\mathbb Z}G)$$ .     

Extension of isomorphisms in abelian p-groups

It is proved that every isomorphism between any two subgroups of a group G retaining the height of the elements in G is extended to an automorphism of the group itself in the class of abelian p-groups without elements of infinite height if and only if G is a closed group with finite Ulam invariants.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 543–548, October, 1973.The author is grateful to I. Kh. Bekker for aid and supervision in performing this research.     

有限超特殊p-群的一个注记

本文获得了以下结果：设G为有限超特殊P-群．则下列条件等价：（1）G的非平凡特征子群的阶相同；（2）G的非平凡特征子群唯一；（3）当p〉2时，exp G=p；当P=2时，G≠D8．     

Finite p-groups all of whose maximal abelian subgroups are soft

A subgroup A of a p-group G is said to be soft in G if CG(A) = A and |NG(A)/A| = p. In this paper we determined finite p-groups all of whose maximal abelian subgroups are soft; see Theorem A and Proposition 2.4.     

Homotopy equivalent group representations and Picard groups op the burnside ring and the character ring

We are concerned with the homotopy theory of group representations and its relation to character theory and the theory of the Burnside ring. We combine the methods of tom Dieck — Petrie [4] and torn Dieck [3] to show that the canonical map from the J-group jO(G), a subquotient of the representation ring RO(G), into the Picard group of the rational representation ring is injective for p-groups G. Moreover we compute the order of the cokernel of this map. We show that the Picard group of the rational representation ring is a direct summand in the Picard group of the Burnside ring. Finally we compute the Picard groups if G is abelian and indicate a computation for general G.     

关于广义超特殊p-群的自同构群

用如下的方式确定了广义超特殊p-群G的自同构群.设|G|=p2n+m,|ζG|=pm,|N|=pl并且G'≤N≤ζG,其中n≥1且m≥2.AutnG表示AutG中平凡地作用在N上的所有自同构形成的正规子群.则(1)当p是奇素数时,AutG/AunG≌Z(p-1)pl-1.进一步地,(i)如果G的幂指数是pm,则Autn...