Tagged Questions

6
votes
2answers
66 views

Embeddings of $GL(n-1,q)$ into $GL(n,q)$

Let $n>2$ and $q$ be a prime power. I want to find all the embeddings of $GL(n-1,q)$ into $GL(n,q)$. An embedding I first thought of is as following. Let $V$ be an $n$-dimensional vector space over ...
3
votes
3answers
111 views

Sylow $p$-subgroups of Finite Matrix Groups

Let $G$ be a subgroup of $GL_n(\mathbb{F}_p)$, with $n\le p$, and let $P$ be a Sylow $p$-subgroup of $G$. Do all non-trivial elements of $P$ have order $p$? I believe the answer is yes, because I ...
3
votes
1answer
93 views

Irreducible subgroups of $\text{GL}(2,p)$

I just wonder any sources that provides the list of all minimal irreducible subgroups of $\text{GL}(2,p)$. Here the term "minimal" means there is no proper subgroup that is irreducible. A list of ...
7
votes
1answer
77 views

Finding the order of the automorphism group of the abelian group of order 8.

So I am given an abelian group of order $8$ such that for all non-identity elements $x^2 = e$ (all elements have order two). So I know the answer is gonna be $168$, but I gotta prove this. So far I ...
2
votes
1answer
93 views

When does there exist a $g\in G$ such that $H = gH'g^{-1}$?

Suppose $H$ and $H'$ are two finite subgroups of $G$. Do there exist theorems that give conditions so that $H$ and $H'$ are conjugate, that is, there exists $g\in G$ such that $H = gH'g^{-1}$? Edit: ...
2
votes
2answers
80 views

Given $v, w$ find a matrix $P$ such that $v = Pw$

How can I show that given two non-zero vectors $v, w \in \mathbb{F}_q^2$ there exists a matrix $P \in SL_2(\mathbb{F}_q)$ such that $v = Pw$?
6
votes
2answers
130 views

Exponent of $GL(n,q)$.

Another exponent problem. $GL(n,q)$ is the group of invertible $n\times n$ matrices over the finite field $GF(q)$, where $q$ is a prime power. I am trying to figure out the exponent of this group. ...
2
votes
0answers
60 views

“unitary group” with respect to non-hermitian matrices?

the group $GU(n,q)$ is usually defined as the group of $n$ by $n$ matrices $X$ over $\mathbb{F}_{q^2}$ such that $X^H X=I_n$, where $I_n$ denotes the identity matrix and $^H$ the hermitian transpose ...
5
votes
1answer
68 views

Is it possible to reverse this sequence of permutations?

Let $ S = (a_1, a_2, ..., a_N) $ be a finite (arbitrarily long) sequence of elements, and let $p_1, p_2, ..., p_n $ be the first $n$ prime numbers, with $n \ge 3$. We apply a sequence of ...
2
votes
1answer
33 views

How can I show that $ASL_n(F)$ is acting 2-transitively?

One of my friends asked me to ask this question here. This is a question from his last exam: Let $$ASL_n(F)=\{T_{A,v}:V_n(F)\to V_n(F)\mid\exists A\in SL_n(F), \exists v\in V_n(F), ...
2
votes
1answer
110 views

Proof that cofactor expansion has unique value

Edit: I'm genuinely not sure why this has gotten little activity. If somebody knows, please tell me, so I can rework it. As a note: I am a purist, and really want to see a proof of this, but I've ...
1
vote
1answer
51 views

Isomorphisms of linear representations of finite groups

Let $G$ be a finite group with representations $\rho_1, \rho_2:G\rightarrow GL(V)$. According to the definition of representation isomorphisms, $\rho_1$ and $\rho_2$ are isomorphic if there exists a ...
0
votes
1answer
46 views

Commutator group of parabolic subgroups of $GL_n(q)$

Let $q$ be a prime power, $G=GL_n(q)$ and $P=q^{km}{:}(GL_k(q) \times GL_m(q))$ be a parabolic subgroup of $G$, where $k+m=n$. What is the commutator group $P'$ of $P$?
3
votes
1answer
139 views

Linear algebra of finite abelian groups

Let $\phi:G \to H$ be a surjective homomorphism of finite abelian groups, and let $g_1, \ldots, g_n$ be an irredundant set of generators (from now on, a basis) for $G$. be a basis for $G$, meaning a ...
1
vote
1answer
95 views

Dimension of subspace fixed by subgroup representation.

If $G$ is an abelian group with cyclic subgroup $H$ and $(\rho,V)$ is a (permutation) representation of $G$. Then I can form a representation of $H$ by considering the composition ...
4
votes
3answers
160 views

Symmetric group [duplicate]

Possible Duplicate: What kind of “symmetry” is the symmetric group about? Could you tell me please, why Symmetric group is called "symmetric"? I found an example with quadrate, ...
1
vote
0answers
75 views

Sets of independent vectors

So we're working in Z2k, the group of bit-vectors of length k and componentwise addition modulo 2. Now I'm trying to make a function yj=1..?(vi) assigning elements of Z2k to n vertices, such that ...
3
votes
2answers
122 views

Proper subgroups of $PSL(3,3)$ are solvable

There is an exercise in a linear group's book that: prove that every proper subgroup of $PSL(3,3)$ is solvable. How can I prove that? the best.
7
votes
1answer
298 views

On the order of elements of $GL(2,q)$?

There's a particular property of the elements of $GL(2,q)$, the general linear group of $n\times n$ matrices over a finite field of order $q$, that I don't understand. First, I know that the order of ...
0
votes
0answers
135 views

Which dimensions can have groups as basis?

For which n is there a set of $n^2$ LI. $n\times n$ matrices, which form a finite group? I know $n=4$ is possible by forming products of Dirac's gamma matrices. These matrices are the identity plus ...
2
votes
1answer
69 views

Non conjugate $p$-subgroups of $\mathrm{GL}_n(\mathbb Z)$

It is probably not that difficult but I can't find an example of two non-conjugate $p$-subgroups (of same order) of $\mathrm{GL}_n(\mathbb Z)$ ($n>1$).
1
vote
2answers
345 views

What are the finite subgroups of $SU_2(C)$?

Is there any reference which classifies the finite subgroups of $SU_2(C)$ up to conjugacy ? What I know is that lifting the finite subgroups of $SO_3(R)$ by the map $SU_2(C) \rightarrow PSU_2(C)$ ...
1
vote
2answers
185 views

Groups/Linear maps

Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have? $\text{GL}_n(\mathbb{F}_3)$ $\text{SL}_n(\mathbb{F}_3)$ Here GL is the general ...