# Questions tagged [centralisers]

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13
questions

**5**

votes

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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
$p$...