Narrow Results

We develop precise geometric descriptions of the conjugacy class $[x]$ and coconjugation set $C(x,x^{\prime })=\{y\in \overline{W}|yx{y}^{-1}=x^{\prime }\}$ for all elements $x,x^{\prime }$ of any affine Coxeter group $\overline{W}$. The centralizer of $x$ in $\overline{W}$ is the special case $C(x,x)$. The key structure in our description of the conjugacy class $[x]$ is the mod-set ${\text{Mod}}_{\overline{W}}(w)=(w-\text{I})R^{\vee }$, where $w$ is the finite part of $x$ and $R^{\vee }$ is the coroot lattice. The set $C(x,x^{\prime })$ is then described by ${\text{Mod}}_{\overline{W}}(w^{\prime })$ together with the fix-set of $w^{\prime }$, where $w^{\prime }$ is the finite part of $x^{\prime }$. For any element $w$ of the associated finite Weyl group $W$, the mod-set of $w$ is contained in the classical move-set $\text{Mov}(w)=\text{Im}(w-\text{I})$. We prove that the rank of ${\text{Mod}}_{\overline{W}}(w)$ equals the dimension of $\text{Mov}(w)$ and investigate type-by-type the surprisingly subtle structure of ${\text{Mod}}_{\overline{W}}(w)$.

Let ${𝒰}=\left[\right.\begin{array}{cc}\hfill A\hfill & \hfill M\hfill \\ \hfill N\hfill & \hfill B\hfill \end{array}\left]\right.$ be a generalized matrix algebra defined by the Morita context $(A,B,M,N,{\mathrm{\Phi }}_{MN},{\mathrm{\Psi }}_{NM})$ and ${𝒵}({𝒰})$ the center of ${𝒰}$. In this paper, under some certain conditions on ${𝒰}$, we prove that if $F:{𝒰}\to {𝒰}$ is an additive map satisfying $[x,F(y)]=0$ for any $x,y\in {𝒰}$ with $xy=0=yx$, then $F$ has the form $F(x)=\lambda x+\tau (x)$ for all $x\in {𝒰}$, where $\lambda \in {𝒵}({𝒰})$ and $\tau$ is an additive map from ${𝒰}$ into ${𝒵}({𝒰})$. Finally as its applications, we characterize Lie centralizer maps and generalized Lie derivations on ${𝒰}$. Moreover we prove that the similar conclusions remain valid on full matrix algebras, triangular algebras, upper triangular matrix algebras.

A proper subgroup $H$ of a finite group $G$ is called a second maximal subgroup of $G$ if $H$ is a maximal subgroup of every maximal subgroup $M$ in $G$ with $H\le M$. In this paper, we investigate the structure of the finite group $G$ in which $C_G(x)$ is a second maximal subgroup for every non-central element $x$ in $G$, and prove that either $G$ is solvable or $G/Z(G)\cong \mathrm{\text{PSL}}(2,r^n)$, where $r=2$ with $n$ a prime or $r=3$ with $n$ an odd prime.

The centralizer of a group element g is the group of elements commuting with g. Powers and, if existent, roots of g are contained in the centralizer. We call a centralizer trivial if it consists of the integer powers of g, only.

In the group of torus homeomorphisms, we study the centralizer of torus diffeomorphisms of hyperbolic Anosov type. As a result, the centralizer can be calculated, isomorphically, in the much smaller group of affine torus automorphisms. In particular, Anosov diffeomorphisms of the 2-torus with a unique fixed point possess trivial centralizer, up to a trivial involution.

The identification of individual reactors from data on reactor cascades, for example in chemical engineering, is one source of motivation for the study of roots of diffeomorphisms. Another possible source is the study of commuting diffeomorphisms in finite-dimensional spatially or spatio-temporally chaotic systems.

We describe an algorithm due to Eric Swenson for testing quasi-convex subgroups of hyperbolic groups for near malnormality. We go on to discuss some strategies for the practical implementation of this algorithm and consider an example. We then show that the methods of Swenson's algorithm extend to give practical solutions to the conjugacy problem for infinite order elements, and to the normalizer and centralizer problems for subgroups satisfying appropriate conditions. The results are presented here in outline — all the algorithms are described in more detail in the author's PhD thesis (University of Warwick, 2001).

Let G be a non-abelian group and N(G) be the set of conjugacy class sizes of G. In 1980s, Thompson posed the following conjecture: if M is a finite non-abelian simple group, G is a finite group with Z(G) = 1 and N(G) = N(M), then M ≅ G. Here, we prove Thompson's conjecture holds for all almost sporadic simple groups.

For any group G, let denote the set of centralizers of G. We say that a group G is a -group if . We prove that every arbitrary -group is center-by-(finite of order . Also we show that the derived length of a nilpotent -group is at most [n - 1/2] + 1.

For a finite group $G$, let $\omega (G)$ be the maximum size of a set of pairwise noncommuting elements in $G$. In this paper, we give an upper bound of $\omega (G)$ for an arbitrary nilpotent group $G$. As an application of this result, we give a partial answer to Question 2.8 of [A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq.7(2) (2000) 139–146]. Also we compute $\omega (G)$ when $\frac{G}{Z(G)}$ is a Frobenius group. Finally we describe structural properties of all groups $G$ with $\omega (G)=5$.

In this paper, we find the number of the element centralizers of a finite group $G$ such that the central factor of $G$ is the projective special linear group of degree 2 or the Suzuki group. Our results generalize some main results of [Ashrafi and Taeri, On finite groups with a certain number of centralizers, J. Appl. Math. Comput.17 (2005) 217–227; Schmidt, Zentralisatorverbände endlicher Gruppen, Rend. Sem. Mat. Univ. Padova44 (1970) 97–131; Zarrin, On element centralizers in finite groups, Arch. Math.93 (2009) 497–503]. Also, we give an application of these results.

Let $G$ be a group. A set $W$ in $G$ is said to be non-commuting if $xy\ne yx$ for any two distinct elements $x$ and $y$ in $W$. If $\left|W\right|\ge \left|X\right|$ for any other non-commuting set $X$ in $G$, then $W$ is said to be a maximal non-commuting set. In this paper, we obtain the cardinality of a maximal non-commuting set in $2$-generated $p$-groups of class two having finite commutator subgroup.

In this paper, we consider the influence of the number of centralizers of noncentral elements of odd order on the solvability of finite groups. Also, we characterize some finite groups in which the centralizer of every noncentral element of odd order is abelian.

Let $R$ be a noncommutative division ring with center $Z$, which is algebraic, that is, $R$ is an algebraic algebra over the field $Z$. Let $f$ be an antiautomorphism of $R$ such that (i) ${[f(x),x^{m(x)}]}_{n(x)}=0$, all $x\in R$, where $m(x)$ and $n(x)$ are positive integers depending on $x$. If, further, $f$ has finite order, it was shown in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] that $f$ is commuting, that is, $[f(x),x]=0$, all $x\in R$. Posed in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] is the question which asks as to whether the finite order requirement on $f$ can be dropped. We provide here an affirmative answer to the question. The second major result of this paper is concerned with a nonnecessarily algebraic division ring $R$ with an antiautomorphism $f$ satisfying the stronger condition (ii) ${[f(x),x^m]}_n=0$, all $x\in R$, where $m$ and $n$ are fixed positive integers. It was shown in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036] that if, further, $f$ has finite order then $f$ is commuting. We show here, that again the finite order assumption on $f$ can be lifted answering thus in the affirmative the open question (see Question 2.11 in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036]).

For any group $G$, let $\mathrm{\text{nacent}}(G)$ denote the set of all non-abelian centralizers of $G$. Jafarian Amiri and Rostami, Groups with a few nonabelian centralizers, Publ. Math. Debrecen87(3–4) (2015) 429–437 put forward the following question: Let $H$ and $G$ be finite simple groups. Is it true that if $|\mathrm{\text{nacent}}(H)|=|\mathrm{\text{nacent}}(G)|$, then $G$ is isomorphic to $H$? In this paper, among other things, we give a negative answer to this question.

Let ${𝒰}$ be a 2-torsion free unital generalized matrix algebra, and $\varphi :{𝒰}\to {𝒰}$ be a linear map satisfying

Let $G$ be a finite AC-group such that $[G:Z(G)]=2^{n}s$, where $n\ge 3$, $s\ge 1$ and $s$ is odd. We prove that if $G$ has $2^{n+1}$ centralizers of elements, then $s=3$, $n$ is an even integer, the set of the sizes of the conjugacy classes of $G$ is $\{1,2^n,3\cdot 2^{n-1}\}$ and $G/Z(G)$ is a Frobenius group whose Frobenius kernel is an elementary abelian $2$-group of order $2^n$ and the Frobenius complement is a group of order $3$.

In this paper, we introduce a notion of pseudo-multidegree function on non-commutative rings having Poincaré–Birkhoff–Witt (PBW) bases with the aim of establishing a criterion to determine whether the centralizer of an element has finite dimension over the non-commutative ring. We illustrate our results with remarkable families of algebras studied in the literature.

The aim of this paper is to use the graphical properties of a group mapping to study the algebraic properties of its centralizers. Equivalent conditions for the simplicity of the centralizer nearring of a given group mapping are given. Examples are provided to demonstrate and delimit our results.

In this paper we show that some finiteness properties on a centralizer of a particular subgroup can be inherited by the whole group. Among other things, we prove the following characterization of polycyclic groups: a soluble group G is polycyclic if and only if it contains a finitely generated subgroup H, formed by bounded left Engel elements, whose centralizer C_G(H) is polycyclic. In the context of Černikov groups we obtain a more general result: a radical group is a Černikov group if and only if it contains a finitely generated subgroup, formed by left Engel elements, whose centralizer is a Černikov group. The aforementioned results generalize a theorem by Onishchuk and Zaĭtsev about the centralizer of a finitely generated subgroup in a nilpotent group.

For a non-abelian group G, we associate the non-commuting graph ∇ (G) whose vertex set is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In this paper, we prove that Aut(J₂) and Aut(M^cL) are characterized by their non-commuting graphs.

The concept of an I-matrix in the full 2 × 2 matrix ring M₂(R/I), where R is an arbitrary UFD and I is a nonzero ideal in R, is introduced. We obtain a concrete description of the centralizer of an I-matrix in M₂(R/I) as the sum of two subrings 𝒮₁ and 𝒮₂ of M₂(R/I), where 𝒮₁ is the image (under the natural epimorphism from M₂(R) to M₂(R/I)) of the centralizer in M₂(R) of a pre-image of , and the entries in 𝒮₂ are intersections of certain annihilators of elements arising from the entries of . It turns out that if R is a PID, then every matrix in M₂(R/I) is an I-matrix. However, this is not the case if R is a UFD in general. Moreover, for every factor ring R/I with zero divisors and every n ≥ 3, there is a matrix for which the mentioned concrete description is not valid.