All Questions
36 questions
4
votes
1
answer
441
views
Large(ish) finite non-abelian subgroups of $\operatorname{GL}_n \mathbb C$ for $n>70$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\SU{SU}\newcommand{\C}{\mathbb{C}}$My question is about large order finite non-abelian subgroups of $\GL_n\C$ without an ...
- 1,150
3
votes
1
answer
268
views
Finite-maximal subgroups of orthogonal groups
I define a finite subgroup $H$ of a group $G$, finite-maximal if for any $g\in G\setminus H$, $\langle H,g\rangle$ is infinite.
My question is now to find the finite-maximal subgroups of $\mathrm{SO}...
- 245
14
votes
0
answers
527
views
Is the monster group maximal in SO(196883)?
$\DeclareMathOperator\SO{SO}$The smallest degree of a nontrivial complex representation of the monster group $ M $ is $ 196883 $. This irrep has Schur indicator $ 1 $, so the image must lie in the ...
- 4,081
2
votes
0
answers
65
views
Are the integer points of a simple linear algebraic group 2-generated?
Set Up:
Let $ K $ be a totally real number field. Let $ \mathcal{O}_K $ be the ring of integers of $ K $. Let $ G $ be a simple linear algebraic group. Suppose that $ G(\mathbb{R}) $ is a compact Lie ...
- 4,081
10
votes
2
answers
914
views
Finite subgroups of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$
Question 1:Is there a reference that lists all possible finite subgroups and their orders of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$ for $n=4$ or even higher $n$ over the real numbers?
I can only find ...
- 26.5k
4
votes
0
answers
134
views
Finite subgroups of $\mathrm{SL}_2(\mathbb{O})$
What are the finite subgroups (subloops that are groups) of $\mathrm{SL}_2(\mathbb{O})$? In particular, are there any not contained in a $\mathrm{SL}_2(\mathbb{H})$?
- 2,773
5
votes
0
answers
217
views
Almost conjugate subgroups of compact simple Lie groups
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$Let $G$ be a compact connected Lie group.
Definition:
Two finite subgroups $H_1,H_2$ of $G$ are said to be almost ...
- 2,446
4
votes
1
answer
237
views
Aschbacher classes for compact simple group
Posted this to MSE several weeks ago and it got 3 upvotes but no answers or even comments so I'm cross-posting to MO
Aschbacher's theorem says that every maximal subgroup of a finite simple classical ...
- 4,081
1
vote
1
answer
383
views
$ S_4 $ subgroups and $ \operatorname{SO}_3(\mathbb{R}) $
$\DeclareMathOperator\SO{SO}$I posted this on MSE 10 days ago and it got 3 upvotes but no answers or comments, so I'm cross-posting to MO.
Background: The group of rotations $ \SO_3(\mathbb{R}) $ has ...
- 4,081
6
votes
1
answer
567
views
Finite simple groups and $ \operatorname{SU}_n $
A follow-up question to Alternating subgroups of $\mathrm{SU}_n $.
$\DeclareMathOperator\PU{PU}\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\PSL{PSL}$Let $ \PU_m $ be the projective unitary group, ...
- 4,081
11
votes
1
answer
357
views
Alternating subgroups of $\mathrm{SU}_n $
$\DeclareMathOperator\PSU{PSU}$Let $ \PSU_n $ be the projective unitary group. Let $ A_m $ be the alternating group on $ m $ letters.
$ A_5 $ is a maximal closed subgroup of $ PSU_2 \cong SO_3(\mathbb{...
- 4,081
2
votes
1
answer
365
views
Self-normalizing implies maximal for subgroup of compact Lie group
Consider the compact group $ G=\operatorname{SO}_3(\mathbb{R}) $. The closed subgroups of $ G $ (other than the trivial group 1 and the whole group $ G $) are exactly $ O_2$, $\operatorname{SO}_2 $ ...
- 4,081
0
votes
1
answer
130
views
Number of reduced decompositions of the dihedral group $D_6$ [closed]
The Weyl group of $\frak{g}_2$ is the dihedral $D_6$. Let us denote its longest element by $w_0$. How many reduced decompositions does $w_0$ have?
- 327
5
votes
1
answer
508
views
Finite maximal closed subgroups of Lie groups
Cross-posted from MSE
https://math.stackexchange.com/questions/4272017/finite-maximal-closed-subgroups-of-lie-groups
$\newcommand{\G}{\mathcal{G}} \newcommand{\K}{\mathcal{K}} \DeclareMathOperator\SU{...
- 4,081
3
votes
1
answer
503
views
Is the representation of finite simple groups fully understood?
Is the representation of finite simple groups fully understood? To clarify, I mean have all the simple representations (even finite dimensional) been classified in terms of some classifying set, such ...
- 203
7
votes
2
answers
1k
views
Finite subgroups of $\operatorname{U}(2)$
Famously, the finite subgroups of $\operatorname{SU}(2)$ admit an ADE classification.
Question. Is there a similar result for finite subgroups of $\operatorname{U}(2)$? Are they
classified? If this ...
- 489
2
votes
0
answers
164
views
Explicit tensor product decomposition for the representations of PSL(2,q)
$\DeclareMathOperator\PSL{PSL}$Let the type of the character theory of a finite group $G$ be the list $[[d_1,n_1], \dotsc, [d_k,n_k]]$ with $1=d_1 < \dotsb < d_k$ and $n_i$ the number of ...
3
votes
0
answers
71
views
Conjugacy classes in reductive group under adjoint action of parabolic subgroup
Given a reductive group $G$ over a finite field and a parabolic subgroup $P$ , I wonder what are the orbits in $G$ under the adjoint action of $P$. This should be standard, but I can only find results ...
- 1,846
8
votes
1
answer
208
views
Can the defining rep of $E_7$ split over a finite subgroup while the adjoint remains simple?
Does the (simply connected compact) Lie group $E_7$ contain a finite subgroup $G \subset E_7$ such that the $56$-dimensional irrep of $E_7$ splits over $G$ as $28 \oplus \overline{28}$, but the $133$-...
- 54.6k
2
votes
0
answers
200
views
What are the points about representation of groups? [closed]
For a fixed (let say finite-, or Lie-, to respect the historical motivations) group, why does the study of all its linear representations over a fixed field, leads to some knowledge about its ...
- 41
2
votes
0
answers
127
views
Multiplicative subgroups of $GL(V)$ which are almost additively closed
Edit:
According to comments of YCor and Vincent, I revise the question.I appreciate their comments:
Let $G$ be a group.
We say that $G$ is a special group if it is isomorphic to a subgroup $H$ ...
- 356
15
votes
1
answer
784
views
The completion of the space of finite groups
Edit: I revise the question based on the comment conversations
Let $\mathcal{F}$ be the set of all equivalence classes of finite groups under the "Isomorphism" equivalence relation.
We define ...
- 356
4
votes
1
answer
633
views
Homomorphisms from binary polyhedral group to compact Lie groups
Are homomorphisms from binary polyhedral groups to (simple and simply connected) compact Lie groups classified?
For cyclic groups, the result is well known (see e.g. Kac's "Infinite dimensional Lie ...
- 6,123
3
votes
0
answers
128
views
Extension of Tits' theorem on groups with a BN-pair of rank ≥ 3
Tits has proved that a finite simple group $G$ with a BN-pair of rank $n \ge 3$, is of Lie type. Let $B$ be the Borel subgroup and $(W,S)$ the Coxeter system. The subset lattice of the set $S$ is ...
4
votes
0
answers
226
views
Possible dimensions for triples of unitary irreducible representations whose tensor product contains the identity
For which triples $\{A,B,C\}$ of positive integers does there exist a (finite or compact) group $G$ with unitary irreducible representations of dimensions $A$,$B$, and $ C$ whose tensor product ...
- 163
1
vote
1
answer
167
views
Finite groups normalizing a torus
Let $G$ be a semi-simple linear algebraic group over the complex numbers, e.g. the special linear group. Can you find an example of a finite sub-group $H$ of $G$ which does not normalize any maximal ...
- 2,384
5
votes
1
answer
2k
views
Weyl groups of $E_6$ and $E_7$
The Weyl group $W_6$ of the Lie algebra $E_6$ is of order 51840, the automorphism group of the unique simple group of order 25920, while the Weyl group $W_7$ of the Lie algebra $E_7$ is of order ...
- 951
8
votes
0
answers
666
views
Approximating Lie groups by finite groups
How can one approximate compact Lie groups by finite groups?
My wish is something like this:
Let $G$ be a compact Lie group.
There is a sequence of nested finite subgroups $G_n$ so that $G_n\to G$...
- 8,086
5
votes
1
answer
549
views
Special linear groups contained in symplectic groups
Let $q$ be a power of prime $p$, and $n, m, k$ positive integers such that $mk=2n$ and $2\leq m<2n$. Let $\mathrm{Sp}(2n,q)$ be the symplectic group of dimension $2n$ over $\mathrm{GF}(q)$ and $\...
- 533
0
votes
0
answers
110
views
Brauer characters of finite simple group $E_8(5)$
I would like to find the irreducible characters of the group $E_8(5)$ (mod 2)?
Can anyone help? (I am elementary in working with Brauer characters)
Many thanks
- 169
1
vote
1
answer
274
views
The compact Lie group contains a finite subgroup $\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \mathbb{Z}_{n_3}$
Given a finite Abelian group: $G=\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \mathbb{Z}_{n_3}$, where ${n_1},{n_2},{n_3}$ are arbitrary positive integers. ${n_1},{n_2},{n_3}$ may have or may not ...
- 10.5k
8
votes
2
answers
2k
views
The number of conjugacy classes of the simple group PSL(2,q)
If $q=p^a$ , where $p$ is a prime number, then I would like to know the number of conjugacy classes related to elements of order $p$ and $2$ in the simple group $PSL(2,q)$ .
- 169
1
vote
1
answer
104
views
The action of graph automorphism of finite symplectic group on maximal subgroups
Let $G=Sp(4,2^f)$ with $f>1$. Based on the facts when $f$ is small, I would feel the following:
$G$ has two conjugacy classes of subgroups isomorphic to $SO^+(4,2^f)$. One is in Aschbacher's class ...
- 767
11
votes
2
answers
1k
views
Finite subgroups of $PGL(3,K)$
It is well-known that finite subgroups of $PGL_2(\mathbb{C})$ are cyclic groups, dihedral groups, A4, S4 and A5 and each of these groups occurs exactly once (up to conjugacy). These facts are ...
- 125
2
votes
0
answers
123
views
Finite subgroups (lattices) in the large N limit of SU(N)
I would like to gain some information about the discrete subgroups (lattices) of SU(N) Lie groups. I have already read some answers and references concerning the N=3 and N=4 cases. I am more ...
- 21
3
votes
0
answers
572
views
How to find the normalizer of a finite subroup in a Lie group?
If a group $G$ is generated by finitely many subgroups $G_i$ and $H$ a subgroup of $G$, under which conditions can $N_G(K)$, the normalizer of $K$ in $G$, be generated by all the normailizers $N_{G_i}(...
- 31