All Questions
Tagged with finite-groups modular-representation-theory
12 questions with no upvoted or accepted answers
24
votes
0
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813
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Revising the proof of CFSG
This is an oft-quoted excerpt from John Thompson's article "Finite Non-Solvable Groups":
“... the classification of finite simple groups is an exercise in taxonomy. This is
obvious to the ...
- 1,433
12
votes
0
answers
340
views
Does every finite group have a small projective representation (over some ring)?
Question. Let $G$ be a finite group. Can we find some (commutative) ring $R$ and some positive integer $d=O(\log\lvert G\rvert)$ such that $G$ can be found as a subgroup of $\operatorname{PGL}_d(R)$?
...
- 602
12
votes
0
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420
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Non-isomorphic groups with same character tables and different Brauer character tables
Let $A$ be a discrete valuation ring with perfect residue field $k$ of characteristic $p$ and field of fractions $K$ of characteristic $0$. Let $G$ and $H$ be two finite groups and assume that $K$ is ...
6
votes
0
answers
266
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What is the Fourier transform in modular representation theory?
For a finite group $G$ there is the Fourier transform $\displaystyle \hat{f}(\rho)=\sum_{g \in G} f(g)\rho(g)$ with inverse $$\displaystyle f(g)=\frac{1}{|G|}\sum_{\rho}d_{\rho}\operatorname{Tr}\left(\...
- 571
5
votes
0
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205
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Can modular representation theory be used to prove Sylow's existence theorem?
Edit 20/12: I added a more precise question at the bottom of the post.
Given a finite group $G$ and a prime $p$, we want to prove that $G$ has a $p$-subgroup $P$ such that $|G:P|$ is not divisible by $...
- 1,433
5
votes
0
answers
130
views
subgroups of $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose mod-2 image is the symmetric group
Let $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ be a closed subgroup of the symplectic group over the $2$-adic integers whose image under the mod-$2$ homomorphism $\pi : \mathrm{Sp}_{2g}(\mathbb{Z}_2)...
- 1,298
3
votes
0
answers
109
views
Exact structures on representations of a finite group
For simplicity assume $G$ is a (finite) $p$-group, and $k$ is field of characteristic $p$, so that there exists a unique simple $kG$-module the trivial module $k$. I am looking for a class of short ...
- 542
3
votes
0
answers
133
views
Isomorphism of certain irreducible representations over finite fields
We are given a faithful representation of a cyclic group of order 5 $\rho: C_5=G \rightarrow End_{\mathbb{F}_3}(V) $ with $dim_{\mathbb{F}_3}V=8$ as vector space. It is also known that $V=U\oplus W$ ...
- 269
2
votes
0
answers
57
views
Brauer pairs associated to a normalizer subsystem in the fusion system of a block of a finite group
Let $G$ be a finite group and let $k$ be an algebraically closed field of positive characteristic $p$. Let $b$ be a block of $kG$ and let $(P,e)$ be a maximal $(G,b)$-Brauer pair. For every subgroup $...
- 81
2
votes
0
answers
116
views
Loewy structure of $S_4$
How to deduce the Loewy Structure of $kS_4$ where $k$ has characteristic 2. I can compute the Cartan matrix and Decomposition matrix with Brauer Characters without difficulties. But when it comes to ...
- 51
1
vote
0
answers
129
views
$G$-invariant bilinear maps
Let $G$ be a finite group and $M$ a finitely generated $\mathbb{Z}G$-module. Then $M \otimes k$ is a representation of $G$ for any field $k$. I am interested in the number of ways we can turn $M \...
1
vote
0
answers
54
views
Largest almost quasisimple group that acts on a spin module
I'd like to know the largest almost quasisimple group $G$ which acts faithfully and irreducibly on the spin module for an odd dimensional orthogonal group $\Omega(2k+1,q)$, or on each of the two half-...
- 121