Questions tagged [centralisers]

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5
votes
0answers
165 views

Centralizer of elements in the upper-triangular matrices

Let $p$ be a prime number and $G=\operatorname{GL}_n ( \mathbb{Z} / p \mathbb{Z} )$ such that $n\leq p$. Consider the set $U$ of upper-triangular matrices of $G$ having entries of $1$ on the diagonal. ...
3
votes
0answers
61 views

Bounds on the component group of the center of a reductive group arising as centralizer

Let $k$ be an algebraically closed field and let $G$ be a reductive, but not necessarily connected, algebraic group over $k$. (For me: reductive group = no non-trivial connected normal unipotent ...
6
votes
2answers
229 views

Centralizer of a cyclic subgroup within the group algebra $\mathbb{C} S_N$ of the symmetric group

Let us take the group algebra $\mathbb{C} S_N$ and the subgroup $H=Z_N$ generated by the element $\sigma=(123\dots N)$, which is a cyclic shift. What is the structure of the centralizer of $H$ within $...
1
vote
1answer
155 views

Centralizers in Jacobson-Witt Lie algebras

Recall the (Jacobson-)Witt Lie algebras in positive characteristic: $W(n,1)$ is the Lie algebra of derivations of $\Bbbk[X_1,\dots,X_n]/(X_1^p,\dots,X_n^p)$. (For simplicity; more generally, I'm ...
1
vote
0answers
48 views

A variation on Dixmier's counterexample concerning centralizers in $A_1$

This question asks the following: "Suppose $k$ is a field of characteristic zero and $P$ and $Q$ are commuting elements of the first Weyl algebra. Is it true that $P$ and $Q$ are polynomials in some ...
2
votes
1answer
282 views

Connectedness of centralizers of semisimple Lie-algebra elements under the action of a semisimple algebraic group

Let $\newcommand{\GG}{\mathbf{G}}\newcommand{\g}{\mathfrak{g}}\GG$ be a connected semisimple algebraic group over the algebraically closed field $k=\overline{\mathbb{F}_q}$, and let $\g$ be its Lie-...
0
votes
0answers
102 views

Connected unipotent groups acting on an affine variety (re: stabilizers)

Let $U$ be a connected unipotent algebraic group over a field of characteristic $p>0$. Assume $U$ acts on an affine variety $X$ by regular maps. Is it true that the stabilizers of rational points ...
2
votes
1answer
146 views

Discrete subgroup of centralizer of transvections in isometries acts properly discontinuously

My question will rely on a clarification of a proof, which I simply don't understand. Let us denote by $X$ a pseudo-riemannian symmetric space and define $$ Z_{\mathrm{Iso}\left(X\right)}G(X) = \{\, ...
4
votes
1answer
469 views

A question on conjugacy classes of central involutions in a finite group

An involution $a$ of a group $G$ is called central if there exists a sylow $2$-subgroup $H$ of $G$ such that $a \in C_G(H)$, or equivalently if the centralizer of $a$ contains a sylow $2$-subgroup. ...
3
votes
1answer
556 views

Involution centralizers in simple groups

I often see lower bounds on the size of centralizers of involutions in finite (nonabelian) simple groups, but is there a general upper bound for the size of an involution centralizer in such a ...
6
votes
1answer
549 views

Centralizers of elements in general linear group over Z mod prime power

I would like to know for which elements $x$ in $G:=Gl_n(\mathbb{Z}/\ell^e\mathbb{Z})$ their centralizers $C_G(x):=\{ y \in G \mid xy=yx\}$ are abelian groups. Here, $n$ is an integer $\geq 2$ and $\...
0
votes
0answers
154 views

Centralizers of elementary abelian subgroups of $p$-groups

Let $P$ be a $p$-group. It is known that if $E$ is a maximal elementary abelian subgroup of rank 2 in $P$, then $C_P(E)/E$ is cyclic where $C_P(E)$ denotes the centralizer of $E$ in $P$. This is ...
6
votes
2answers
659 views

Lie algebras and non-smoothness of centralisers in bad characteristic

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p>0$. For $x\in G$, let $C_{G}(x)$ denote the centraliser, considered as a group scheme over $k$. If $p$...