Finite Groups of Bounded Rank with an Almost Regular Automorphism of Prime Order

We prove that if a finite group G of rank r admits an automorphism of prime order having exactly m fixed points, then G has a -invariant subgroup of (r,m)-bounded index which is nilpotent of r-bounded class (Theorem 1). Thus, for automorphisms of prime order the previous results of Shalev, Khukhro, and Jaikin-Zapirain are strengthened. The proof rests, in particular, on a result about regular automorphisms of Lie rings (Theorem 3). The general case reduces modulo available results to the case of finite p-groups. For reduction to Lie rings powerful p-groups are also used. For them a useful fact is proved which allows us to glue together nilpotency classes of factors of certain normal series (Theorem 2).     

Groups Admitting an Automorphism of Prime Order with Finite Centralizer

We consider a group G with an automorphism of finite, usually prime, order. If G has finite Hirsch number, and also if G satisfies various stronger rank restrictions, we study the consequences and equivalent hypotheses of having only finitely many fixed-points. In particular we prove that if a group G with finite Hirsch number ${\mathfrak{h}}$ admits an automorphism ${\varphi}$ of prime order p such that ${\vert C_{G}(\varphi) \vert = n < \infty,}$ then G has a subgroup of finite index bounded in terms of p, n and ${\mathfrak{h}}$ that is nilpotent of p-bounded class.     

一类不能作为自同构群的奇阶群

本文考虑如下问题：怎样的有限群可以作为另一个有限群的全自同构群？我们首先证明，若有限群Ｋ有一个正规Ｓｙｌｏｗｐ－子群使得｜Ｋ：Ｚ（Ｋ）｜ｐ＝ｐ２，那么Ｋ有２阶自同构．利用这个结果，我们证明了，若奇阶群Ｇ具有阶Ｐｓｍ（１≤ｓ≤３），ｐ为｜Ｇ｜的最小素因子，ｐｍ，ｍ无立方因子，则Ｇ不可能作为全自同构群．     

Central Automorphisms and Inner Automorphisms in Finitely Generated Groups

Let G be a group and Aut_c(G) be the group of all central automorphisms of G. We know that in a finite p-group G, Aut_c(G) = Inn(G) if and only if Z(G) = G′ and Z(G) is cyclic. But we shown that we cannot extend this result for infinite groups. In fact, there exist finitely generated nilpotent groups of class 2 in which G′ =Z(G) is infinite cyclic and Inn(G) < C* = Aut_c(G). In this article, we characterize all finitely generated groups G for which the equality Aut_c(G) = Inn(G) holds.     

有限生成的幂零群的共轭分离性质

研究了有限生成的幂零群中元素的共轭分离问题.设ω表示全部素数组成的集合,π是ω的非空真子集,G是有限生成的幂零群,则下述三条等价:(i)如果x和y是G中的任意两个不共轭的元素,则x和y在G的某个有限p-商群中不共轭,其中p∈π;(ii)如果x和y是G中的任意两个不共轭的元素,则x和y在G的某个有限π-商群中不共轭;(iii)G的挠子群T(G)是π-群且G/T(G)是Abel群.同时举例说明:设G是有限生成的无挠幂零群,对于任意素数p,x和y都在G的有限p-商群G/G~p中共轭,但x和y在G中不共轭.     

Conjugacy Separability of Descending HNN-Extensions of Finitely Generated Abelian Groups

It is proved that an arbitrary descending HNN-extension of a finitely generated Abelian group is conjugacy separable.     

On Certain Automorphism Groups of Finitely Generated Groups

Mathematical Notes -     

二秩无扭群的自同构群和只有两个自同构的二秩无扭群

本文利用Ｋｕｒǒｓ不变量理论和不定方程理论，讨论了二秩无扭群的自同构群，以及有零高元的二秩无扭群只有两个自同构的充要条件．     

Frattini子群是无限循环群的有限生成幂零群

完整地确定了Frattini子群是无限循环群的有限生成幂零群的结构,证明了下面的定理.设G是有限生成幂零群,则G的Frattini子群是无限循环群当且仅当G可以分解为G=S×F×T,其中F是秩为s的自由Abel群,T=Z_m_1⊕Zm_2⊕…⊕Z_m_u,m_1,m_2,…,m_u都是大于1的没有平方因子的自然数,m_1|m_2|…|m_u,■式中d_1,d_2,…,d_r都是正整数,d_1|d_2|…|d_r.进一步,(d_1,d2,…,d_r;s;m_1…,m_2,…,m_u)是群G的同构不变量,即若群H也是Frattini子群是无限循环群的有限生成幂零群,那么G同构于H的充要条件是它们有相同的不变量.     

无平方因子阶群的自同构群阶的上确界

群阶为素数方幂（即ｐ－ 群）时已得到该群自同构群阶的上确界,而对于其他情形的群,同样的问题要复杂得多． 本文在群阶无平方因子且为偶数时,给出了这类群的自同构群阶的上确界．     

Suzuki-Ree群的自同构群的阶分量刻画

在《数学学报》2013年第56卷第4期中,"Suzuki-Ree群的自同构群的一个新刻画"一文证明了Aut(~2F_4(q)),q=2~f和Aut(~2G_2(q)),q=3~f,可由其阶分量刻画,其中f=3~s,s为正整数.本文证明了Aut(~2B_2(q)),q=2~f和Aut(2G2(q)),q=3~f,也可由其阶分量刻画,其中f为奇素数.结合二者得到结论:Suzuki-Ree单群的所有的素图不连通的自同构群皆可由其阶分量刻画.     

2pq阶群的自同构群

继续「１～２」等的工作,给出了２ｐｑ阶群Ｇ的自同构群ＡｕｔＧ的准确构造。     

Direct Products with Abelian Automorphism Groups

The article considers when the direct product of two finite groups has an Abelian automorphism group.     

广义四元数群的Coleman外自同构群

在这篇注记中,我们利用群的射影极限性质证明了广义四元数群的Coleman外自同构群或者是1或者是一个初等阿贝尔2-群.     

树的自同构群的一个新的恒等式

通过计数给定基础图的标根地图数，本文得到了树的自同构群的如下恒等式∑T∈T(n)Πd∈D(T)^(d-1)！/|AutT|=（2n-1)!/n!(n 1)!，这里T（n),D(T)分别表示n阶不同构树集，树T的次序列。     

Automorphism Groups of Computably Enumerable Predicates

We study automorphism groups of two important predicates in computability theory: the predicate W_y and the graph of a universal partially computable function. It is shown that all automorphisms of the predicates in question are computable. The actions of the automorphism groups on some index sets are examined, and we establish a number of results on the structure of these. We also look into homomorphisms of the two predicates. In this case the situation changes: all homomorphisms of the universal function are computable, but in each Turing degree, homomorphisms of W_y exist.     

不充当自同构群的有限群

设ｐ为奇素数，本文将用一些新的技巧来证明，当Ｐ是阶小于Ｐ＾１１的交换Ｐ－群时，自同构群方程Ａｕｔ（Ｘ）＝Ｐ无解。这个结果使ＭａｃｈＨａｌｅ在１９８３年的工作得到了突破，并且我们所给的方法具有广泛性。     

On Orbits of Automorphism Groups

If G is a finite group and if A is a group of automorphisms of G whose fixed point subgroup is C _G (A) then every subgroup F of C _G (A) acts on the set of orbits of A in G. The peculiarities of this action are used here to derive several results on the number of orbits of A in an economical manner.Original Russian Text Copyright © 2005 Deaconescu M. and Walls G. L.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 533–537, May–June, 2005.     

充当自同构群的有限群的几点注记

设p为奇素数,本文给出了中心循环中心商的阶小于p^5的有限p-群的完全分类并且给出它们中无对合自同构的群的自同构群的阶。由此,我们找到了能作为有限群自同构群的p^mq^n阶群和p^n阶群,统一和推广了Curran在1988年和Caranti与Scoppola在1990年的文章的所有结果。     

具有交换幂零根基的完备李代数的自同构群

给出了复数域上具有交换幂零根基的完备李代数的自同构群．