All Questions
Tagged with linear-groups group-theory
158 questions
1
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0
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25
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Why a $\Gamma$ for the general semiaffine group in its notation?
This question is about notation.
I often see the general semiaffine group of degree $n$ over a field $K$ written as $A\Gamma L(n,K)$, $\Gamma A(n,K)$, $A\Gamma L_n(K)$, or $\Gamma A_n(K)$...
Why the ...
- 3,990
1
vote
0
answers
39
views
How to Construct a Larger Set of Generators for Sp(2n, q) to Optimize Diameter? [closed]
For the symplectic group Sp(2n, q) over a finite field of order q, I am interested in constructing a set of generators that produces a Cayley graph with a "good diameter." Specifically:
By &...
- 19
0
votes
0
answers
47
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Is the Valentiner Group isomorphic to PGL(3, 4)?
In this answer to a question from a while back, it says that the Valentiner group is isomorphic to $PGL_3(\mathbb{F}_4)$. However, when I implement in sage:
...
1
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0
answers
31
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The order of a linear profinite group over commutative profinite ring
Recall that one can define the order of a profinite group using the notion of supernatural number. Let $A$ be a commutative profinite ring and let $G$ be a topologically finitely generated closed ...
- 585
1
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1
answer
63
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No continuous isomorphism from $O_4$ to $O_{3,1}$
I am hoping to show that there is no continuous isomorphism from the orthogonal group $O_4$ to the Lorentz group $O_{3,1}$. My approach so far is to make use of the fact that a matrix $A$ is in $O_{3,...
- 152
3
votes
1
answer
123
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The rank of Sylow subgroup of special linear groups over finite fields
Let $p,\ell$ be two primes, and let $\mathbb{F}_{q}$ be a finite field of order $q=p^r$. We define the rank of a finite group $G$ to the smallest cardinality of a generating set for $G$. We denote by $...
- 585
1
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0
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43
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Condition on finitely generated subgroup of $GL_2(\mathbb{Q})$ to be free
I am considering finitely generated subgroup $G=\langle A,B,C\rangle$ of $GL_2(\mathbb{Q})$ such that $A,B,C$ all have the upper triangular form
$$A=\begin{pmatrix}a_1 & a_2 \\ 0 & a_3\end{...
- 1,355
2
votes
1
answer
71
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A generalization of Baumslag-Solitar groups
I am wondering about the following generalization of the group $B_{1,2}=\langle a,b\, |\, bab^{-1}=a^2\rangle$:
$$
G_k=\langle a_1,a_2,\ldots,a_{k+1}\, |\, a_{i+1}a_ia_{i+1}^{-1}=a_i^2, i=1,2,\ldots,k\...
- 427
1
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1
answer
84
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The order of $H = \{X \in \mathrm{GL}_2(\mathbb{Z}_p) \mid \det(X) = 1\}$ with the definition
Let be a prime $p$ and $H = \{X \in \mathrm{GL}_2(\mathbb{Z}_p) \mid \det(X) = 1\}$. I know that the order of a element $g \in \mathrm{GL}_2(\mathbb{Z}_p)$ is the less $k$ such that $g^k = e$, but I ...
- 175
1
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1
answer
135
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Generators for ${\rm SL}_2(\mathbb{Z}/8\mathbb{Z})$ [closed]
Let ${\rm SL}_2(\mathbb{Z}/8\mathbb{Z})$ be the special linear group over the finite ring $\mathbb{Z}/8\mathbb{Z}$. By How to calculate |SL2(Z/NZ)|
, we know that the order of the group ${\rm SL}_2(\...
- 585
2
votes
1
answer
117
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Prove that every finite subgroup of $SL_2(\mathbb{C})$ is conjugate to a finite subgroup of $SU_2$
I want to prove this proposition.
For any finite subgroup $G$ of $SL_2(\mathbb{C})$, I showed that $G$ is unitary with respect to an inner product that we define properly.
But I couldn't find the ...
- 31
2
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0
answers
40
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Showing that two orbit spaces are isomorphic
Consider the following two group actions:
$O_n\times Sym_n(\mathbb{R})\rightarrow Sym_n(\mathbb{R})$
Given by $(A,B)\mapsto ABA^{-1}$, where $O_n$ denotes the group of orthogonal matrices and $Sym_n(\...
- 167
4
votes
0
answers
125
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Proving certain triangle groups are infinite
[Cross-posted to MathOverflow]
Consider the Von Dyck group
$$ G = \langle x,y\mid x^a=y^b=(xy)^c=1\rangle $$
where $a,b,c\ge3$. Because $G$ is infinite and residually finite, it has an infinite family ...
- 5,465
5
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1
answer
254
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When is general linear group isomorphic to special linear group?
Can $\operatorname{GL}_n(\Bbb F_{q})$ be isomorphic to $\operatorname{SL}_m(\Bbb F_{r})$, where $\Bbb F_q$, $\Bbb F_r$ are finite fields with $q,r$ elements respectively?
By considering the center, $$\...
- 643
4
votes
1
answer
129
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Prove that $\operatorname{GL}_n(k)$ and $\operatorname{SL}_n(k)$ cannot be symmetric group?
Problem. Prove that $\operatorname{GL}_n(k)$ and $\operatorname{SL}_n(k)$ cannot be isomorphic to $S_m$, $m\geq 4$ if $k$ is a finite field with at least two elements.
I am trying to argue by looking ...
- 643
3
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0
answers
86
views
Number of homomorphisms from $\Bbb Z_p$ to $\operatorname{GL}_2(\Bbb Z_q)$?
I am trying to count the total possible number of group homomorphisms from $\Bbb Z_p$ to $\operatorname{GL}_2(\Bbb Z_q)$, where $p<q$ are primes and $\Bbb Z_p$ denotes the additive group modulo $p$....
3
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1
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84
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Semilinear subgroup $\Gamma L(1,2^n)$ of group $GL(n,2)$
Let $\phi$ be an automorphizm of $\mathbb{F}_{2^n}$ and $a \in \mathbb{F}_{2^n}$.
Let's consider a set of functions $\Gamma L(1,2^n) = \{g_{a,\phi} | a\in\mathbb{F}_{2^n}, \phi \in Aut(\mathbb{F}_{2^n}...
- 301
-1
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1
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55
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why square matrix is group for multipy
There are 4 requirements in order to be group
for $$(R^{n*n},*), n\in{N}$$
Closer $ \forall{A,B}\in{R^{n*n}}:A*B\in{R^{n*n}}$
Associativity $\forall{A,B,C}\in{R^{n*n}}:A*(B*C)=(A*B)*C$
Neutral ...
- 67
0
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0
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47
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Hyperplanes in $SL_2(\mathbb{R})$ containing conjugacy classes of matrices
Let $SL_2(K)$ be the special linear group of rank 2 over a field $K$, as an affine group scheme cut out by the equation $ad - bc = 1$ in $\mathbb{A}^4_K$.
Let $H$ be an arbitrary, homogeneous, ...
- 1,280
4
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1
answer
171
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Are $S^1$ and $O_2(\mathbb{R})$ isomorphic?
Let $S^1 = \{z \in \mathbb{C} | |z| = 1\}$ and $O_2(\mathbb{R}) = \{A \in GL_2(\mathbb{R})|A^TA = I_2 \}$. Is the group $(S^1, \cdot)$ isomorphic to $(O_2(\mathbb{R}), \cdot)$? Prove it.
This is a ...
4
votes
1
answer
336
views
Why can the Orthogonal group be split up in this way?
In my groups course at university, we’ve spent a while on the orthogonal group, leading up the the conclusion that
$$
\mathrm{O}_n
=
\mathrm{SO}_n
\mathbin{\dot{\cup}}
\begin{pmatrix}
-1 ...
- 43
-1
votes
1
answer
65
views
Generation a subgroup of $GL_2(\mathbb{C})$
Let $G$ be a group and let $S$ be a nonempty subset of a group $G$.
The subgroup of $G$ generated by $S$, which is denoted by $\langle S\rangle$,
is equal to the set of all elements of $G$ that can be ...
- 67
3
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1
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178
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Does a given subset of an infinite group generate a finite or infinite subgroup?
Consider a fixed infinite group, such as $SL(n, \mathbb{R})$. Let $S$ be a given finite subset of the elements of $SL(n, \mathbb{R})$, further suppose that $S$ is closed under inverses. Is there any ...
- 93
1
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0
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403
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Prove that the quaternion group of order 8 is isomorphic to a subgroup of $SL_2(\mathbb{F_3})$ generated by the following two elements.
The Problem: Prove that the subgroup of $SL_2(\mathbb{F_3})$ generated by $\begin{pmatrix}0&-1\\1&0\\\end{pmatrix}$ and $\begin{pmatrix}1&1\\1&-1\\\end{pmatrix}$ is isomorphic to the ...
- 1,487
2
votes
1
answer
208
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The general linear group $GL(n, \mathbb{C})$ has no proper subgroup of finite index.
Problem: Show that the general linear group $G = GL(n, \mathbb{C})$ has no proper subgroup of finite index.
I wrote down a proof, but not quite sure if it is right, especially about the part about ...
- 1,103
2
votes
1
answer
366
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7.5.4 of Stillwell Naive Lie Theory
In section 7.5 of John Stillwell's 'Naive Lie Theory' he constructs a proof of what he calls the 'Tangent space visibility' theorem: that for any path-connected matrix Lie group $G$ with a discrete ...
4
votes
1
answer
268
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If every finite quotient group of a finitely generated linear group G is solvable, then G is solvable
For this question, I was able to show that each finite quotient is polycyclic: Suppose $N$ is a normal subgroup of finite index. Then, all subgroups of $G/N$ are finite, so $G/N$ is Noetherian. A ...
- 535
1
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0
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168
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If every finitely generated subgroup of a linear group G is solvable, then G is solvable
I've been working on this question for a long time now and I still have no idea how to approach it. I tried induction on the dimension of the matrices in G but can't complete the induction step. Any ...
- 535
0
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0
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61
views
Is it true that $F^*\wr S_n$ is a solvable group, when $n\leq 5$?
Let $U$ be a linear space over a division ring $D$, $G_1$ a subgroup of
$GL(U)$, and $\Gamma$ a subgroup of a symmetric group $S_k$ on $\{1,\ldots,k\}, k>1$.
The cartesian product $U^k=V_1$ can ...
4
votes
2
answers
199
views
Finding the normalizer of $\left\{ \left(\begin{matrix} x &0 \\0 & y \end{matrix}\right) : x,y\in \mathbb R\setminus\{0\} \right\}$
I'm having some trouble with the following question:
Let $G=\text{GL}_2(\mathbb R)$. What are the elements of the set: $$N_G \left( \underbrace{\left\{
\left(\begin{matrix} x &0 \\0 & y \end{...
- 5,109
3
votes
0
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208
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What are the subgroups of the projective general linear group of degree 3 over a finite field?
I'm investigating something and the question of what the subgroups of $ \text{PGL}(3,\mathbb{F}_p)$ are has come up. I've found this Group Properties Wiki page, but it doesn't contain much useful ...
- 29
2
votes
1
answer
181
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The Sylow $2$-subgroups of $SL(2,q)$ are not cyclic for $q \equiv 3$ mod $4$?
I'm reading through the free version of the book Finite Groups from Daniel Gorenstein and theorem 8.3 on page 42 caught my attention. I understand the proof so far, but the second page of the proof is ...
- 121
2
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0
answers
104
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Group action on one-dimensional subspaces
Assume $p=(q^n-1)/(q-1)$ with $p$ prime and let $V:=\mathbb F^n_q$ denote an $n$-dimensional vector space over $\mathbb F_q$. There is a natural action of $PSL(n,q)$ on the $(q^n-1)/(q-1)$ one-...
2
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1
answer
588
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Finding an element of order $p+1$ in $\text{SL}_2(\mathbb{F}_p)$ and Sylow subgroups of $\text{SL}_2(\mathbb{F}_p)$
Let $p$ be a prime number. I am trying to show that $\text{SL}_2(\mathbb{F_p})$ has an order $p+1$ element and use this to show that for every odd prime $q$, the $q$-Sylow subgroup of $\text{SL}_2(\...
- 645
2
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0
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401
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Order of $PSL(n,q)$
Being $F$ the field of order $q$ that linear groups are defined here, there is something I can't understand.
I know that $|SL(n,q)| = \dfrac{|GL(n,q)|}{(q-1)}$, and I know that $PSL(n,q) = SL(n,q)/Z(...
- 515
2
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1
answer
145
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why do matrices in orthogonal group have determinant 1 in characteristic 2?
Let $q$ be quadratic form on $V=K^n$ where $K$ is a field in characteristic $2$.
Let $b(x,y)=q(x+y)-q(x)-q(y)$ be the bilinear form associated to $q$ and set $V^\perp=\{x\in V\mid b(x,y)=0 \text{ for ...
- 21
-1
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1
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263
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Show that $GL(2, \Bbb Q)$ is isomorphic to a subgroup of $GL(3, \Bbb Q)$ [closed]
What should be my approach for this particular question and general approach to prove isomorphism of one group to another ?
9
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2
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201
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Does $G$ being a subgroup of $GL(n, \mathbb Z/p\mathbb Z)$ for all odd primes $p$ imply it is a subgroup of $GL(n, \mathbb{Z})$?
It is known that any finite subgroup of $GL(n, \mathbb{Z})$ is isomorphic to a subgroup of $GL(n, \mathbb Z/p\mathbb Z)$ for any odd prime $p$ (see here). I am wondering if there is a converse to to ...
- 2,748
2
votes
0
answers
21
views
Reference for $SO_Q$ maximal in $SL_n$
I would like to find a reference for the following fact, which is sort of "well known":
Let $Q$ be a positive definite non-degenerate quadratic form in $n$ variables over an algebraically ...
- 71
4
votes
2
answers
613
views
Explicit isomorphism between ${\rm SL}(2,{\Bbb R})$ and ${\rm SU}(1,1)$
I have heard that there exists an isomorphism of real algebraic groups as in the title. I am asking for an explicit isomorphism.
Motivation: I need such an isomorphism for a calculation of Galois ...
- 1,226
0
votes
0
answers
41
views
Generalizations of linear groups
There are quite many general results about finitely generated linear groups (e.g. the Malcev theorem that they are residually finite, the Tits alternative for them, etc.). Since at least in the real ...
- 636
0
votes
1
answer
97
views
Show that a complex, compact, abelian linear group is finite (using Liouville's theorem)
I'm trying to solve the following problem from the second chapter of Rossmann's book on Lie groups:
This seems like a very interesting result (and a cool application of Liouville's theorem), but I'm ...
- 4,458
3
votes
2
answers
963
views
Why is $SL(n, \mathbb{R})$ the kernel of $\det : GL(n, \mathbb{R}) \mapsto\Bbb R^*$?
The special linear group of invertible matrices is defined as the kernel of the determinant of the map:
$$\det:GL(n,\mathbb{R}) \mapsto \mathbb{R}^*$$
In my mind the kernel of a linear map is the set ...
- 769
1
vote
1
answer
231
views
Book about linear groups
Please advise a good book about linear groups, their actions (as, for example, PGl acts on projective space, etc.), as well as the relationship between themselves.
(ideally if they were considered ...
- 464
1
vote
2
answers
2k
views
SU(1,1) isomorphic to SL(2,R), but SU(2) is not
I am surprised by the fact that $\mathrm{SU}(1, 1)$ group is isomorphic to $\mathrm{SL}(2, \mathbb{R})$, but $\mathrm{SU}(2)$ is not isomorphic to $\mathrm{SL}(2, \mathbb{R})$.
The first statement is ...
- 113
2
votes
2
answers
442
views
Confused by quotient group (whats the operation): Show quotient group $GL_n(K)/SL_n(K)$ is abelian.
In my introductory abstract algebra course, the quotient group $G/H$ was defined as
$$G/H=\{gH:g\in G\}$$
which is a set of sets. In an exercise, I should show that for the group of invertible ...
- 2,289
2
votes
1
answer
70
views
Group of $2\times2$ matrices that are isomorphic to $SU(2)$
Studying physics I encountered the $SU(2)$ group, in the context in which I use it $SU(2)$ is the group of the $2\times2$ unitary matrices with determinant equal to one. Out of curiosity, I searched ...
- 333
2
votes
0
answers
506
views
Generators of $SL_2 (F_q)$
I am working on the following problem.
Let $\mathbb{F}_q$ be a finite field with $q \neq 9$ elements and $a$ be a generator of the cyclic group $\mathbb{F}_q^{\times}$. Show that $\mathrm{SL}_2(\...
- 71
1
vote
1
answer
314
views
Small order examples of non-nilpotent finite groups in which every minimal normal subgroup intersects the center nontrivially
I know that $p$-groups have the property, and all nilpotent groups also do. I wanted to have some non-nilpotent examples. Then I checked quasisimple groups, and found every proper normal subgroups of ...
user517681
-1
votes
1
answer
416
views
conjugacy classes of the special orthogonal group $SO(2)$ [closed]
I'm doing some research in Group Theory and have come across conjugacy classes. In general I can determine the conjugacy classes for most groups.
However the conjugacy classes of the special ...
user741289