All Questions
Tagged with classical-groups group-theory
42 questions
3
votes
1
answer
58
views
Finding generators of Aschbacher subgroups
Aschbacher famously showed that the maximal subgroups of the finite classical groups can be split into various categories. How does one find explicit generators for these groups, when the field is ...
- 403
-1
votes
0
answers
38
views
Locating the maximal subgroups when the character table is known
The character table for many families of classical groups, such as $SL_3(q)$, are known.
The maximal subgroups are also had for many of the classical groups, found according to their Aschbacher ...
- 403
2
votes
1
answer
53
views
Derived subgroup of finite symplectic groups
Let $G=\text{Sp}_{2n}(q)$ be a symplectic group, where $q$ is a prime-power. Is it true that $G'=G$?
This isn't true in a few small cases, e.g. $\text{Sp}_4(2)\cong S_6$.
I think we can use Aschbacher'...
- 750
3
votes
1
answer
142
views
Show Sp(1) is the same as SU(2)
I'm reading John Stillwell's Naive Lie Theory but got stuck at a result in section 3.4 when he began introducing the symplectic groups.
He defined the symplectic groups as the following:
On the space $...
- 51
1
vote
1
answer
68
views
An explicit isomorphism between $\mathbb{R}^+ \times {\rm Spin}^c(3,1)$ and ${\rm GL}^+(2,\mathbb{R})\times {\rm GL}^+(2,\mathbb{R})$? [closed]
I am interested in the following isomorphism
$$
\begin{align}
\mathbb{R}^+\times {\rm Spin}^c(3,1)& \cong \mathbb{R}^+\times {\rm Spin}(3,1) \times {\rm U}(1) \tag{1}\\
&\cong \mathbb{C}^+\...
- 2,649
0
votes
2
answers
148
views
Stabiliser group [closed]
Let $G=\operatorname{Sp}_{2r}(2)$. There are two orbits of $G$ on the natural $G$-set, one having the identity, the other having all the remaining elements. What is the subgroup of $G$ that stabilizes ...
- 1,321
2
votes
1
answer
106
views
Conjugate subgroups by permutation matrices
Is the following statement true or false?
In $G=\operatorname{GL}_{n}(\textbf{C})$, two elementary abelian $2$-subgroups $X$ and $Y$ of $G$ whose generators are all diagonal matrices are conjugate if ...
- 1,321
2
votes
0
answers
78
views
Intersection of identity components
Let $e_{1}$ and $e_{2}$ be involutions in the algebraic group $G=\operatorname{PGL}_{n}(\mathbb{C})$. Do we have $$C_{G}(\langle e_{1},e_{2}\rangle)^{\circ} = C_{G}(e_{1})^{\circ}\cap C_{G}(e_{2})^{\...
- 1,321
4
votes
1
answer
160
views
What is the Order Of Symplectic Group(4,2) and Symplectic Group(4,3) from the Classical groups of Atlas?
I have been working on Symplectic group of classical groups. I am trying to find Sylow-2 Subgroups of Symplectic Group(4,2) and Symplectic Grouop(4,3) through GAP, I am facing a problem regarding ...
0
votes
0
answers
55
views
$N/C$ contains $\operatorname{GL}_{2}(2)$
In $G = \operatorname{PGL}_{4}(\mathbb{C})$, a subgroup $E = <a,b,c,d>$ where $$a =\Delta
(\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix})_{2}, b =\Delta
(\begin{pmatrix}
0 & 1\\
1 &...
- 1,321
2
votes
1
answer
163
views
Linear group contained in symplectic group
Do we have SL$(n,2) \leqslant$ Sp$(2n, 2)$? And better yet, $\operatorname{SL}(n,q) \leqslant$ Sp$(2n, q)$ or
$\operatorname{GL}(n,q) \leqslant$ Sp$(2n, q)$?
I checked using Magma for small degrees ...
- 1,321
1
vote
0
answers
63
views
Matrix group inclusion
It is true that
$Sp_{2}(2)$ is a subgroup of the matrix group $\begin{pmatrix}
Sp_2(2) & 0\\
*_{1\times 2} & 1\\
\end{pmatrix}$
where $*_{1\times2}$ denotes a $1\times 2$ matrix with ...
- 1,321
1
vote
0
answers
35
views
Construction of $SL(2,5)^4$ in a maximal subgroup in Magma
There is a Class $2$ maximal subgroup $SL(2,5)^{4}.4^{3}.S_{4}$ (denoted by $MM_2$ in the following code) in $PGL(8,5)$. I am trying to locate the $SL(2,5)^4$ part in Magma. But after I constructed ...
- 1,321
4
votes
1
answer
79
views
construct the normaliser of a subgroup and then construct the subgroup
There is a maximal nontoral (not contained in a conjugate of a fixed maximal torus) elementary abelian $2$-subgroup of rank 6 in $G = PGL(8,7)$. I denote this group by $A$. Its normaliser $N_{G}(A)$ ...
- 1,321
1
vote
1
answer
88
views
split maximal torus construction
In PGL(n,q) there is a split maximal torus T of order $(q-1)^{n-1}$. How to construct this in Magma?
Let's use the example of $PGL(4,11)$. I took a detour to construct it:
...
- 1,321
3
votes
1
answer
55
views
inclusion among finite symplectic groups
This might be a silly question, but do we have the following?
$Sp(2,q) < Sp(4,q) < Sp(6,q) < Sp(8,q) <....$
I checked the list of maximal subgroups of $Sp(n,q)$ for $n = 4,6,8,10,12$ and ...
- 1,321
2
votes
0
answers
48
views
Generation of $SU(n,q^2)$ by subgroups isomorphic to $SU(2,q^2)$
Throughout, let $q$ be an odd prime power. Let $GF(q^2)$ be the field with $q^2$ elements.
My question concerns the generation of the special unitary group $SU(n,q^2)$, where $n \ge 2$, by certain ...
- 21
20
votes
1
answer
408
views
Does $SL_2(K) \simeq SL_2(L)$ imply $K\simeq L$?
Let $K$ and $L$ be two fields. Assume characteristics are not 2. I can show in a quite elementary way that if the statement $SL_2(K) \simeq SL_2(L) \implies K \simeq L$ holds, then for $n \geq 2$, the ...
- 607
0
votes
1
answer
569
views
Results on Maximal subgroups of $\text{GL}(n,q)$
Consider, $\text{GL}(n,q)$, which is the group of $n\times n$ invertible matrices with entries from the finite fields $\mathbb{F}_q$, with $q$ elements.(We know $q$ is a prime-power). My question is ...
- 4,175
1
vote
1
answer
173
views
Maximal subgroups of ${\rm PSL}(2,8)$ [duplicate]
I’m reading a note on the subgroup structure of classical subgroups. According to Corollary 2.2 (d), since $8=2^3$ is a prime power of even number $2$, ${\rm PSL}(2,8)$ has a maximal subgroup ${\rm ...
user818044
4
votes
0
answers
548
views
Understanding Conjugacy classes of the Unitary group over finite fields
Consider the General linear group $GL(n,q)$ over the finite field $\mathbb{F}_q$ of $q$ elements. The unitary group $U(n,q)$ is described as a subgroup of $GL(n,q^2)$ which is the set of linear maps ...
- 4,175
2
votes
0
answers
48
views
Intersection of isometric subspaces in a space over a finite field
Suppose $V$ is a symplectic, unitary, or orthogonal space over a finite field $\mathbb{F}_q$ of characteristic $p$.
Suppose $H\subseteq U\subseteq W$ is a chain of subspaces of $V$, where $\dim U=\...
- 687
4
votes
1
answer
272
views
Is $PGL(n,\Bbb{R}) \cong SL(n+1, \Bbb{R})$ for even n?
Is the following claim correct?
Claim: If n is even, $PGL(n,\Bbb{R}) \cong SL(n+1, \Bbb{R})$.
Proof:
Recall $PGL(n,\Bbb{R}) \cong GL(n+1, \Bbb{R})/Z$, where $Z = \{M | M=\alpha I, \alpha \in \Bbb{...
- 456
3
votes
1
answer
227
views
Transitivity of $\Omega$ in subspace actions
Let $G=O(V, Q)$ be a finite orthogonal group acting naturally on a space $V\cong \mathbb{F}_q^n$ equipped with a quadratic form $Q$. Assume $n=\dim V\geq c$ for some large enough constant $c$ in order ...
- 687
3
votes
1
answer
73
views
Reference request: structure of stabilisers of totally isotropic subspaces in orthogonal (and unitary) groups
I am looking for a book or paper which covers the structure of stabilisers in
$GO(n,F)$, $SO(n,F)$ (or maybe in $\Omega(n,F)$) of totally isotropic subspaces of dimension $k$. Can you please suggest ...
- 727
2
votes
1
answer
96
views
Sylow subgroup of a subgroup 5
I am aiming for looking for all the Sylow subgroups of classical groups, which gives me a seemingly elementary question: what can we say about Sylow subgroups of a subgroup if we know all Sylow ...
- 984
1
vote
1
answer
129
views
Does $\text{SU}(1,1)$ act transitively on $\mathbb S^1=\{z\in\mathbb C\mid |z|=1\}$?
Let $\text{SU}(1,1)=\left\{\left[ \begin{array}{ccc}
\alpha & \beta \\\overline\beta & \overline\alpha \end{array} \right]\mid \alpha,\beta\in \mathbb C,|\alpha|^2-|\beta|^2=1\right\}$ and $\...
- 12.7k
4
votes
1
answer
278
views
Hall subgroups of $\mathrm{PSL}$
The following is an exercise in Peter Cameron's notes on classical groups.
Exercise 2.10 (a) Show that $\mathrm{PSL}(2,5)$ fails to have a Hall subgroup of some admissible order.
(b) Show that $\...
- 2,886
0
votes
1
answer
52
views
Kernel of ${\rm GL}(n,F)$ on ${\rm PG}(n-1,F)$ over a division ring $F$
I am reading Peter Cameron's note on Classical Groups and I got confused with Proposition 2.1 on page 14.
I have no problem in proving that the elements in kernel are scalars. However, I don't ...
- 4,495
5
votes
0
answers
124
views
Centralizer of $U(n)$ inside $U(nm)$
Let $n$ and $m$ be two positive integers. There is a canonical inclusion $U(n) \rightarrow U(nm)$ given by the tensor product with the unit matrix $\mathbf{1}_m$.
What is the centralizer of $U(n)$ ...
- 1,715
1
vote
0
answers
108
views
Centraliser of orthogonal group in general linear group
I'm looking for a reference that tells me either the centraliser of $\mathrm{O}^\epsilon(n,q)$ in $\mathrm{GL}(n,q)$, or how to describe the module homomorphisms of $\mathrm{O}^\epsilon(n,q)$ acting ...
- 488
1
vote
0
answers
188
views
Maximal (permutation) subgroups of $PSL(2,p)$
I have been trying to look at the maximal subgroups of $\mathrm{PSL}(2,p)$ for $p > 2$ prime and believe I have a sufficiently good idea of how those isomorphic to $C_p \rtimes C_{\frac{1}{2}(p-1)}$...
- 90
2
votes
0
answers
57
views
Eigenspaces of semisimple element in finite unitary group
Let $G=GU(V)=GU_{n}(q)$ - the general unitary group acting on a vector space $V$, over the finite field of $q^2$ elements.
Let $1 \neq s\in G$ be semisimple and $\lambda \in \mathbb{F}_{q^2}$ an ...
- 300
2
votes
2
answers
192
views
Finding the spinor norm of an element using a proposition in 'The Maximal Subgroups of the Low-Dimensional Finite Classical Groups'.
Let $F=\mathbb{F}_{q}$, where $q$ is an odd prime power. Let $e,f,d$ be a standard basis for the $3$-dimensional orthogonal space $V$, i.e. $(e,e)=(f,f)=(e,d)=(f,d)$ and $(e,f)=(d,d)=1$. I have an ...
- 387
1
vote
0
answers
34
views
Subgroup structure description
I've been reading a paper on group theory, and have come across this description of a subgroup of the special linear group:
Let $G=SL(d,q)$ and $H< G$ with $H \cong (SL(k,q) \times SL(d-k,q)).(q-...
- 11
6
votes
1
answer
218
views
Which finite simple groups contain $PSL(2,q)$ for some $q\geq 4$?
Which nonabelian finite simple groups contain $PSL(2,q)$ for some $q$?
Obviously $PSL(2,q)$ themselves do. Also, as $PSL(2,4)\cong PSL(2,5)\cong A_5\subset A_n,\; n\geq 5$, alternating (nonabelian ...
- 21.1k
1
vote
1
answer
224
views
An automorphism that is not inner.
Consider the group $G=SL_3(\mathbb{C})$. I want to show that the automorphism $\phi$ of $G$ given by $\phi(x)=(x^{-1})^T$ is not inner. Probably I should do this by contradiction, i can show that if $\...
- 23
1
vote
1
answer
99
views
Relation between general linear groups for subfield $K$ of $F$ of finite index.
If $F$ is a field and $d \ge 1$.
Let $K$ be a subfield of $F$ with finite index $k = [F : K]$. Then $F$ is a $k$-dimensional vector space over $K$. Thus every $F$-vector space is also a $K$-vector ...
- 18.4k
0
votes
1
answer
182
views
Centralizer of element in group PSL(2,F_p)
Is it true, that $\forall g\in PSL(2,F_p)\setminus\{e\}$, $Z(g)$ is Abelian?
I think that this is true, but i can't find a simple proof.
user60882
1
vote
0
answers
142
views
Non-central proper normal subgroups of unitary groups over fields
Short version: Can someone give an example of an anisotropic Hermitian form over a field such that its corresponding projective unitary group is not simple?
Let $F$ be a (commutative, associative, ...
- 56.4k
2
votes
0
answers
52
views
Centralizing a maximal flag in a symplectic group
Short version: I'm confused about maximal totally-isotropic flags versus maximal flags: do they have the same centralizer in the classical group?
Let $F$ be a field, $V$ be a finite dimensional ...
- 56.4k
5
votes
2
answers
192
views
Determining if these surjections have sections
Let $\pi:\ \operatorname{GL}(2,k)\ \longrightarrow\ \operatorname{PGL}(2,k)$ be the canonical homomorphism, and pick some finite subgroup $G\subset \operatorname{PGL}(2,k)$. Then we have an exact ...
- 21.1k