On automorphisms of some finite <Emphasis Type="Italic">p</Emphasis>-groups

We give a sufficient condition on a finite p-group G of nilpotency class 2 so that Aut _c (G) = Inn(G), where Aut _c (G) and Inn(G) denote the group of all class preserving automorphisms and inner automorphisms of G respectively. Next we prove that if G and H are two isoclinic finite groups (in the sense of P. Hall), then Aut _c (G) ≃ Aut _c (H). Finally we study class preserving automorphisms of groups of order p ⁵, p an odd prime and prove that Aut _c (G) = Inn(G) for all the groups G of order p ⁵ except two isoclinism families.     

On Commuting Automorphisms of p-Groups

Let G be a group. If the set 𝒜(G) = {α ∈Aut(G) | xα(x) = α(x)x, for all x ∈ G} forms a subgroup of Aut(G), then G is called 𝒜(G)-group. We show that the minimum order of a non-𝒜(G) p-group is p ⁵ for any prime p. We also find the smallest group order of a non-𝒜(G) group. This is related to a question introduced by Deaconescu, Silberberg, and Walls [4 Deaconescu , M. , Silberberg , Gh. , Walls , G. ( 2002 ). On commuting automorphisms of groups . Arch. Math 79 : 423 – 429 .[Crossref] , [Google Scholar]]. Moreover, we prove that for any prime p and for all integer n ≥ 5, there exists a non-𝒜(G) group of order p ⁿ .     

On Finite n-Acentralizer Groups

Let G be a group and Aut(G) be the group of automorphisms of G. Then the Acentralizer of an automorphism α ∈Aut(G) in G is defined as C _G (α) = {g ∈ G∣α(g) = g}. For a finite group G, let Acent(G) = {C _G (α)∣α ∈Aut(G)}. Then for any natural number n, we say that G is n-Acentralizer group if |Acent(G)| =n. We show that for any natural number n, there exists a finite n-Acentralizer group and determine the structure of finite n-Acentralizer groups for n ≤ 5.     

Automorphisms of Hyperbolic Groups and Graphs of Groups

Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups G_e are rigid (i.e. Out(G_e) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.     

Group Theoretic Properties of the Group of Computable Automorphisms of a Countable Dense Linear Order

We compare Aut(Q), the classical automorphism group of a countable dense linear order, with Aut _c (Q), the group of all computable automorphisms of such an order. They have a number of similarities, including the facts that every element of each group is a commutator and each group has exactly three nontrivial normal subgroups. However, the standard proofs of these facts in Aut(Q) do not work for Aut _c (Q). Also, Aut(Q) has three fundamental properties which fail in Aut _c (Q): it is divisible, every element is a commutator of itself with some other element, and two elements are conjugate if and only if they have isomorphic orbital structures.     

On central automorphisms that fix the centre elementwise

Let G be a group and let Aut _c (G) be the group of central automorphisms of G. Let be the set of all central automorphisms of G fixing Z(G) elementwise. In this paper we prove that if G is a finite p-group, then = Inn(G) if and only if G is abelian or G is nilpotent of class 2 and Z(G) is cyclic. This work was supported in part by the Center of Excellence for Mathematics, University of Isfahan, Iran. Received: 30 October 2006     

Noncyclic Graph of a Group

We associate a graph Γ _G to a nonlocally cyclic group G (called the noncyclic graph of G) as follows: take G\ Cyc(G) as vertex set, where Cyc(G) = {x ? G| 〈x, y〉 is cyclic for all y ? G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of Γ _G is finite if and only if Γ _G has no infinite clique. We prove that if G is a finite nilpotent group and H is a group with Γ _G  ? Γ _H and |Cyc(G)| = |Cyc(H)| = 1, then H is a finite nilpotent group. We give some examples of groups G whose noncyclic graphs are “unique”, i.e., if Γ _G  ? Γ _H for some group H, then G ? H. In view of these examples, we conjecture that every finite nonabelian simple group has a unique noncyclic graph. Also we give some examples of finite noncyclic groups G with the property that if Γ _G  ? Γ _H for some group H, then |G| = |H|. These suggest the question whether the latter property holds for all finite noncyclic groups.     

On the Howson Property of HNN-Extensions with Nilpotent Base Group and Amalgamated Free Products of Nilpotent Groups

We give necessary and sufficient conditions under which an amalgamated free product of finitely generated nilpotent groups is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated). Also we prove that if G = ? t, K | t ^?1 At = B ?, where K is a finitely generated and infinite nilpotent group and A, B non-trivial infinite proper subgroups of K, then G is not a Howson group. The problem of deciding when an ascending HNN-extension of a finitely generated nilpotent group is a Howson group is still open.     

On Infinite Camina Groups

A group G is called a Camina group if G′ ≠ G and each element x ∈ G?G′ satisfies the equation x ^G  = xG′, where x ^G denotes the conjugacy class of x in G. Finite Camina groups were introduced by Alan Camina in 1978, and they had been studied since then by many authors. In this article, we start the study of infinite Camina groups. In particular, we characterize infinite Camina groups with a finite G′ (see Theorem 3.1) and we show that infinite non-abelian finitely generated Camina groups must be nonsolvable (see Theorem 4.3). We also describe locally finite Camina groups, residually finite Camina groups (see Section 3) and some periodic solvable Camina groups (see Section 5).     

Twisted Conjugacy Classes for Polyfree Groups

Let G be a finitely generated polyfree group. If G has nonzero Euler characteristic then we show that Aut(G) has a finite index subgroup in which every automorphism has infinite Reidemeister number. For certain G of length 2, we show that the number of Reidemeister classes of every automorphism is infinite.     

Fitting Height of a Finite Group with a Metabelian Group of Automorphisms

Let M = FH be a finite group that is a product of a normal abelian subgroup F and an abelian subgroup H. Assume that all elements in M?F have prime order p, and F has at most one subgroup of order p. Examples of such groups are dihedral groups for p = 2 and the semidirect product of a cyclic group F by a group H of prime order p such that C _F (H) = 1 or |C _F (H)| =p and C _{F/C F (H)}(H) = 1. Suppose that M acts on a finite group G in such a manner that C _G (F) = 1. We prove that the Fitting height h(G) of G is at most h(C _G (H))+ 1. Moreover, the Fitting series of C _G (H) coincides with the intersection of C _G (H) with the Fitting series of G.     

The Automorphism Group of a Class of Nilpotent Groups with Infinite Cyclic Derived Subgroups

The automorphism group of a class of nilpotent groups with infinite cyclic derived subgroups is determined. Let G be the direct product of a generalized extraspecial Z-group E and a free abelian group A with rank m, where E ={(1 kα_1 kα_2 ··· kα_nα_(n+1) 0 1 0 ··· 0 α_(n+2)...............000...1 α_(2n+1)000...01|αi∈ Z, i = 1, 2,..., 2 n + 1},where k is a positive integer. Let AutG G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G of G, and AutG/ζ G,ζ GG be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζ G of G. Then(i) The extension 1→ Aut_(G') G→ AutG→ Aut(G')→ 1 is split.(ii) Aut_(G') G/Aut_(G/ζ G,ζ G)G≌Sp(2 n, Z) ×(GL(m, Z)■(Z~)m).(iii) Aut_(G/ζ G,ζ GG/Inn G)≌(Z_k)~(2n)⊕(Z)~(2nm).     

On Finite Groups with Few Automorphism Orbits

Denote by ω(G) the number of orbits of the action of Aut(G) on the finite group G. We prove that if G is a finite nonsolvable group in which ω(G) ≤5, then G is isomorphic to one of the groups A₅, A₆, PSL(2, 7), or PSL(2, 8). We also consider the case when ω(G) = 6 and show that, if G is a nonsolvable finite group with ω(G) = 6, then either G ≈ PSL(3, 4) or there exists a characteristic elementary abelian 2-subgroup N of G such that G/N ≈ A₅.     

Automorphisms of Free Left Nilpotent Leibniz Algebras and Fixed Points

ABSTRACT

Let F _m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F _m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F _m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) ^G is not finitely generated.     

Non-Nilpotent Graph of a Group

We associate a graph 𝒩 _G with a group G (called the non-nilpotent graph of G) as follows: take G as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this article, we study the graph theoretical properties of 𝒩 _G and its induced subgraph on G \ nil(G), where nil(G) = {x ∈ G | ? x, y ? is nilpotent for all y ∈ G}. For any finite group G, we prove that 𝒩 _G has either |Z*(G)| or |Z*(G)| +1 connected components, where Z*(G) is the hypercenter of G. We give a new characterization for finite nilpotent groups in terms of the non-nilpotent graph. In fact, we prove that a finite group G is nilpotent if and only if the set of vertex degrees of 𝒩 _G has at most two elements.     

The Group of Outer Automorphisms and the Picard Group of an Algebra

The primary goal of this work is to develop a strategy to compute Pic_K(A) and Out_K(A) where A is a finite-dimensional algebra over a field K. The basic idea is to put the normal subgroup Inn^*(A) of the inner automorphisms of A induced by elements of 1 + J(A), as a common denominator in the fraction Aut_K(A) / Inn(A). The new numerator Aut_K(A)/Inn^*(A) and the new denominator Inn(A)/Inn^*(A) are much easier to deal with than the original Aut_K(A) and Inn(A). The main ingredient to study Out(A) is now the appearance of an Abelian group Ch(, K), the group of acyclic characters of the quiver of A, that can be completely calculated. We show how to apply these results to compute the Picard group of a split finite-dimensional algebra in several cases.     

Weak Cayley table groups III: PSL(2,q)

A weak Cayley table isomorphism is a bijection φ:G→H of groups such that φ(xy)～φ(x)φ(y) for all x,y∈G. Here ～ denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ:G→G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I:G→G,x?x^?1. Let 𝒲₀(G) = ?Aut(G),I?≤𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲₀(G). We show that PSL(2,pⁿ) has trivial weak Cayley table group, where p≥5 is a prime and n≥1.     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.