All Questions
14 questions
6
votes
1
answer
659
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 $\...
- 841
6
votes
0
answers
492
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. ...
- 463
4
votes
1
answer
385
views
Finite groups with bounded centralizers
Let $G$ be a finite group. For each $x\in G$, the centralizer $\mathbf{C}_G(x)$ must contain $\langle x\rangle$.
QUESTION: What are some interesting results of the following form:
Given some bound on $...
- 1,433
4
votes
1
answer
698
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.
...
- 1,197
4
votes
0
answers
186
views
On 2-groups of exponent 4 and class 2
Suppose A is a 2-group with the following properties:
$\lvert A \rvert = t^3$ with $t$ some even power of $2$;
$A$ and $Z(A)$ (the center of $A$) are of exponent $4$;
$\lvert Z(A) \rvert = t$ and $[A,...
- 4,547
3
votes
1
answer
746
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 ...
- 4,547
3
votes
1
answer
203
views
normalizer quotient is $\operatorname{GL}_2(p)$
Let $p$ be a prime and let $w$ be a primitive $p$-th root of unity in $\mathbb{C}$. There is an element $e$ of order $p$ in $G=\operatorname{PGL}_n(\mathbb{C})$ where $n=pk$ and
$$e=\left[\left(\begin{...
- 501
2
votes
0
answers
167
views
Centralizer of PSL in PGL and of SL in GL: reference request
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\PSL{PSL}$Consider the general linear group $\GL(n,q)$ over a finite field with $q$ elements and ...
- 361
1
vote
1
answer
144
views
$|C(E):C(E)\cap C(Z(U))|=1$ or $p$
Lemma 2.4 from Robert Griess' paper "Elementary abelian $p$-subgroups of algebraic groups" states:
(i) Let $T$ be a finite $p$-group whose Frattini subgroup is cyclic and central. Then $T'$ ...
- 501
1
vote
1
answer
202
views
action of the extra-special group
I'm reading a paper which has this line:
A direct computation shows that $P\Omega_8$($\mathbb K$) has an elementary abelian subgroup $X = 2^2$ such that $C_{P\Omega_8(\mathbb K)}(X) = T_4.2^{1+4}_+$. ...
- 43
1
vote
0
answers
90
views
Central-by-cyclic
This is a following-up question of this.
Lemma 2.4 from Robert Griess' paper "Elementary abelian $p$-subgroups of algebraic groups" states:
(i) Let $T$ be a finite $p$-group whose Frattini ...
- 501
1
vote
0
answers
74
views
$C_G(E)= E \times{\rm PGL}_k(q)$
Let $r$ be an odd prime and $q$ a power of a prime $p$ where $r\neq p$.
If $r^m|n$ and $q\equiv1$ (mod $r$), then $r^{1+2m}.{\rm Sp}_{2m}(r)\le{\rm GL}_n(q)$ and $Center(r^{1+2m}.{\rm Sp}_{2m}(r))\le ...
- 501
1
vote
0
answers
60
views
Centralisers of involutions not quasi-isolated
The quasi-isolated elements of the algebraic group $G=\operatorname{PGL}_{n}(\mathbb{C})$ are classified in the paper "quasi-isolated elements in reductive groups" by C. Bonnafe.
Let's focus ...
- 501
0
votes
1
answer
129
views
Intersection of identity components
Let $e_{1}$ and $e_{2}$ be involutions in the algebraic group $G=\operatorname{PGL}_{n}(\mathbb{C})$. Do we have $$C_{G}(\langle e_{1},e_{2}\rangle)^{\circ} = C_{G}(e_{1})^{\circ}\cap C_{G}(e_{2})^{\...
- 501