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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 ...
Ali Taghavi's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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{...
Ian Gershon Teixeira's user avatar
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 ...
Tim's user avatar
  • 125
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 ...
Mare's user avatar
  • 26.5k
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 $PSL(2,q)$ .
Tina's user avatar
  • 169
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 $133$-...
Theo Johnson-Freyd's user avatar
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 $G_n\to G$...
Joonas Ilmavirta's user avatar
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 ...
Federico Carta's user avatar
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, ...
Ian Gershon Teixeira's user avatar
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 ...
Hebe's user avatar
  • 951
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 $\...
Huangjun Zhu's user avatar
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{...
Ian Gershon Teixeira's user avatar
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 ...
emiliocba's user avatar
  • 2,446
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 ...
Fetchinson0234's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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 ...
Yuji Tachikawa's user avatar
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 $\mathrm{SL}_2(\mathbb{H})$?
Daniel Sebald's user avatar
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 ...
Mark's user avatar
  • 163
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 ...
Dick Johnson's user avatar
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}...
Andrea Aveni's user avatar
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 ...
Simon Lentner's user avatar
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 ...
Sebastien Palcoux's user avatar
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}(...
Gang Han's user avatar
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 $ ...
Ian Gershon Teixeira's user avatar
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 ...
Ian Gershon Teixeira's user avatar
2 votes
0 answers
163 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 ...
Sebastien Palcoux's user avatar
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 ...
huurd's user avatar
  • 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 $H$ ...
Ali Taghavi's user avatar
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 ...
Felino's user avatar
  • 21
1 vote
1 answer
166 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 ...
Giulio's user avatar
  • 2,384
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 ...
wonderich's user avatar
  • 10.5k
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 ...
Binzhou Xia's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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?
Martim Pereir's user avatar
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
Tina's user avatar
  • 169