All Questions
21 questions
1
vote
0
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39
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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
views
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:
...
4
votes
0
answers
125
views
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
2
votes
0
answers
104
views
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-...
9
votes
2
answers
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
1
vote
0
answers
225
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Does $SL(2,3)$ have only $4$ normal subgroup?
I'm looking at the special linear group and have a question:
Does $SL(2,3)$ have only $4$ normal subgroup? How to prove the part "only"?
I can see that the two trivial subgroups, the ...
- 343
5
votes
0
answers
114
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Criterion for $PSL(2,q)$
I am looking for a criterion for a group to be isomorphic to $PSL(2,p^k)$ in terms of its Sylow $p$-subgroups.
For example, let $G$ be finite group of order $p^km$ where $p$ is an odd prime not ...
- 4,431
7
votes
1
answer
178
views
$PSL(2,13)$ has no subgroup of prime index.
I want to show that $PSL(2,13)$ has no subgroup of prime index,where $PSL(2,13) = \frac{SL(2,13)}{\brace-I,I}$.
We have the below fact.
《If $G$ be a simple group and $H$ be a subgroup of $G$ such ...
- 650
6
votes
1
answer
56
views
What is the maximal $m$, such that $\mathbb{Z}_2^m \leq GL(n, 2)$?
Is there any closed formula for the function $m(n)$, that is defined as the maximal $m$, such that there is $GL(n, 2)$ has a subgroup isomorphic to $\mathbb{Z}_2^m$?
The only things I know currently, ...
- 15.8k
1
vote
2
answers
46
views
Show that the Quotient Group $Γ_2(p)/Γ_2(p^k)$ is finite.
Let $p$ be a prime number. Denote by $Γ_2(p)$ the multiplicative group of all $2×2$ matrices
$$ x =
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
$$
with elements $a, b, c, d ∈ \...
- 441
2
votes
1
answer
361
views
Exercise 4.7.8 Dixon-Mortimer "Permutation groups".
Suppose that $m$>$1$ is an integer, and $p$ and $r$ are primes such that $r$
divides $p^{m}-1$ but $r$ does not divide $p^k-1$ for $1$$\leq$ $k$ <$m$. Show that $GL_m$$($$p$$)$ has an irreducible ...
- 303
3
votes
2
answers
1k
views
Find a finite generating set for $Gl(n,\mathbb{Z})$
I need to find a finite generating set for $Gl(n,\mathbb{Z})$. I heard somewhere once that this group is generated by the elementary matrices - of course, if I'm going to prove that $GL(n,\mathbb{Z})$ ...
user100463
3
votes
0
answers
105
views
Subgroups of $SL(n,F)$ which contain $B\cap SL(n,F)$
If $B$ is the usual Borel subgroup of $GL(n,F)$, then what are the subgroups of $SL(n,F)$ which contain $B\cap SL(n,F)$? I don't even know where to start! Any help is appreciated! :)
- 41
1
vote
0
answers
68
views
Showing $PSL(2,9)$ doesn't have a subgroup of order $90$
I got stuck in proving that there is not any subgroup of order $90$ in group $PSL(2,9) = SL(2,9)/Z(SL(2,9))$. Can anyone help?
- 123
2
votes
1
answer
719
views
Properties of point stabilizers of $PSL(2, q)$
Let $G$ be a transitive, non-regular finite permutation group such that each non-trivial element fixing some point fixes exactly two points. Suppose that $G \cong PSL(2, q), q > 5$ and $H = G_{\...
- 18.4k
4
votes
1
answer
262
views
Is a nontrivial finite group of order $n$ always isomorphic to a subgroup of $GL_{n-1}(\mathbb{Z})$?
I saw this question on an old qualifying exam:
Let $G$ be a group of order $n\ge2$. Is such a group always isomorphic to a subgroup of $GL_{n-1}(\mathbb{Z})$?
A simpler problem would be to show ...
- 2,167
8
votes
1
answer
328
views
The cardinality of the group $\mathrm{SL}(k,\mathbb{Z}/n\mathbb{Z})$
I want to calculate the cardinality of the group $\mathrm{SL}(k,\mathbb{Z}/n\mathbb{Z})$, $k,n \in \mathbb{N}$. When $n$ is prime, it is easy to calculate, so I want to know the general case.
- 341
4
votes
1
answer
127
views
Showing all p-local subgroups are char p given that all are contained in char p locals
Question: How does one prove that if every $p$-subgroup $U$ is contained as a subnormal subgroup of a characteristic-$p$, radical, local subgroup containing the normalizer of $U$, then the normalizer ...
- 56.4k
2
votes
1
answer
201
views
Why $PGL(2, 9)$ is not isomorphic to $S_6$?
How can I show that $PGL(2,9)$ is not isomorphic to $S_6$?
My primary idea is to compare the size of conjugacy classes of two well-chosen elements in these groups. Is there another simpler approach?
- 941
3
votes
1
answer
780
views
$p$-Sylow subgroups of $\operatorname{GL}(n, \mathbb{F}_{p^k})$
Is there a way to classify all $p$-Sylow subgroups of $GL(n,\, \mathbb{F}_{p^k})$ where $|\mathbb{F}_{p^k}|=p^k$? Clearly the prime $p$ (that is the characteristic of the base field) is a very ...
- 13.7k
4
votes
1
answer
438
views
Semidirect Products of the form $(\mathbb{Z}_p \times \mathbb{Z}_p)\rtimes_{\phi}\operatorname{GL}(2,p)$
What are the different (non-isomorphic) semidirect products $(\mathbb{Z}_p \times \mathbb{Z}_p)\rtimes_{\phi}\operatorname{GL}(2,p)$, when $\phi \colon \operatorname{GL}(2,p)\rightarrow\operatorname{...
user8186