On Certain Automorphism Groups of Finitely Generated Groups

Mathematical Notes -     

Asymptotic Cones of Finitely Generated Groups

Answering a question of Gromov [7], we shall present an exampleof a finitely generated group and two non-principal ultrafiltersA, B such that the asymptotic cones Con_A and Con_B are nothomeomorphic. 1991 Mathematics Subject Classification 20F06,20F32.     

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.     

Collectives of Automata in Finitely Generated Groups

Mathematical Notes - The present paper is devoted to traversing a maze by a collective of automata. This part of automata theory gave rise to a fairly wide range of diverse problems ([1], [2]),...     

Palindromic Width of Finitely Generated Solvable Groups

We investigate the palindromic width of finitely generated solvable groups. We prove that every finitely generated 3-step solvable group has finite palindromic width. More generally, we show the finiteness of the palindromic width for finitely generated abelian-by-nilpotent-by-nilpotent groups. For arbitrary solvable groups of step ≥3, we prove that if G is a finitely generated solvable group that is an extension of an abelian group by a group satisfying the maximal condition for normal subgroups, then the palindromic width of G is finite. We also prove that the palindromic width of ??? with respect to the set of standard generators is 3.     

On Isoperimetric Profiles of Finitely Generated Groups

We find up to multiplicative constant isoperimetric profiles of wreath products and related groups. Those are the first examples where the explicitly calculated asymptotics of isoperimetric profile is neither polynomial nor exponential.     

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.     

On Degrees of Growth of Finitely Generated Groups

We prove that for an arbitrary function ρ of subexponential growth there exists a group G of intermediate growth whose growth function satisfies the inequality v _G,S (n) ? ρ(n) for all n. For every prime p, one can take G to be a p-group; one can also take a torsion-free group G. We also discuss some generalizations of this assertion.     

Abelian Subgroups of Finitely Generated Kleinian Groups are Separable

By a Kleinian group we mean a discrete subgroup of PSL(2, C).We prove that abelian subgroups of finitely generated Kleiniangroups are separable. In other words, if M = H³/ is a hyperbolic3-orbifold, with finitely generated, then abelian subgroupsof are separable in . 1991 Mathematics Subject Classification20E26, 51M10, 57M05.     

Finitely Generated Nilpotent Groups of Infinite Cyclic Commutator Subgroups

The aim of this paper is to determine the structure and to establish the isomorphic invariant of the finitely generated nilpotent group G of infinite cyclic commutator subgroup. Using the structure and invariant of the group which is the central extension of a cyclic group by a free abelian group of finite rank of infinite cyclic center, we provide a decomposition of G as the product of a generalized extraspecial Z-group and its center. By using techniques of lifting isomorphisms of abelian groups and equivalent normal form of the generalized extraspecial Z-groups, we finally obtain the structure and invariants of the group G.     

On Subgroups of Finite Index in Positively Finitely Generated Groups

This paper proves that a subgroup of finite index in a positivelyfinitely generated profinite group has maximal subgroup growthat most n^log(n). In particular such a subgroup cannot be free,answering a question by L. Pyber. 2000 Mathematics Subject Classification20E28, 20P05.     

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.     

无限循环群被有限生成Abel群的中心扩张

设G是无限循环群被有限生成Abel群的中心扩张,T是G的中心ζG的挠子群.如果T的阶与ζG/(G'⊕T)的挠子群的阶互素,那么群G可分解为G=S×F×T,其中S= 这里d_i都是正整数,满足d_1|d_2|…|d_r,F是秩为s的自由Abel群,T是有限Abel群,T=Z_(e_1)⊕Z_(e_2)⊕…⊕Z_e_t,e_11,满足e_1|e_2|…|e_t,并且(d_1,e_t)=1.进一步,(d_1,d_2,…,d_T;s;e_1,e_2,…,e_t)是群G的同构不变量,即若群H也是无限循环群被有限生成Abel群的中心扩张,T_H是ζH的挠子群.如果T_H的阶与ζH/(H'⊕T_H)的挠子群的阶互索,那么G同构于H的充要条件是它们有相同的不变量.显然,这个结果涵盖了有限生成Abel群的结构定理.     

Solvability of Independent Systems of Equations in Finitely Generated Nilpotent Groups

Mathematical Notes - It is proved that the solvability problem for a finite independent system of equations in a finitely generated nilpotent group can effectively be reduced to a similar problem...     

Ascending Hnn Extensions of Finitely Generated Free Groups Are Hopfian

It is shown that every ascending HNN extension of a finitelygenerated free group is Hopfian. An important ingredient inthe proof is that under certain hypotheses on the group H, ifG is an ascending HNN extension of H, then cd(G) = cd(H) + 1.     

Finitary Power Semigroups of Infinite Groups are not Finitely Generated

For a semigroup S, the finitary power semigroup of S, denotedP_f(S), consists of all finite subsets of S under the usual multiplication.The main result of this paper asserts that P_f(G) is not finitelygenerated for any infinite group G. 2000 Mathematics SubjectClassification 20M05 (primary), 20M30, 20F99 (secondary).     

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.     

When Essential Extensions of Finitely Generated Modules are Finitely Generated

For certain classes 𝒞 of R-modules, including singular modules or modules with locally Krull dimensions, it is investigated when every module in 𝒞 with a finitely generated essential submodule is finitely generated. In case 𝒞 = Mod-R, this means E(M)/M is Noetherian for any finitely generated module M_R. Rings R with latter property are studied and shown that they form a class 𝒬 properly between the class of pure semisimple rings and the class of certain max rings. Duo rings in 𝒬 are precisely Artinian rings. If R is a quasi continuous ring in 𝒬 then R ? A ⊕ T where A is a semisimple Artinian ring and T ∈ 𝒬 with Z(T_T) ≤_ess T_T.     

Finitely Generated Quasi-Proximities

In this paper we deal with the proximal properties of relators which are straightforward generalizations of Weil's uniformities. We prove that every finitely generated quasi-proximity is simple and show that a theorem in Kenyon [4] can be set in a more general form.