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Results tagged with reference-request
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user 35416
This tag is used if a reference is needed in a paper or textbook on a specific result.
7
votes
Accepted
Wedderburn decomposition of special linear groups
$\DeclareMathOperator\M{M}\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\End{End}$As has been mentioned in the comments, the question for algebraically closed fields of characteristic $0$ is equiva …
- 13k
16
votes
Accepted
Finite groups with integral character table
There is no complete classification, but some structural results are known. To give you something to search for: such groups are called $\mathbb{Q}$-groups. There is a whole book devoted to their stru …
- 13k
1
vote
Reference for fact about reduction mod $p$ of a representation of a finite group
I don't know of just a citation, but here is a pretty quick way to deduce this from the literature (I imagine that this might be the argument you had in mind, in which case apologies for telling you t …
- 13k
7
votes
Autobiographies of mathematicians
Edward Frenkel's "Love and Math" is a mix of popular maths book, autobiography, and general declaration of love towards mathematics.
10
votes
Accepted
For which finite groups $G$ is every character a virtual permutation character?
No classification of such groups is known. As you say, for every character to be a virtual permutation character, necessary conditions are that
all irreducible characters are $\mathbb{Q}$-valued; eq …
- 13k
6
votes
0
answers
163
views
Generalisation of the Witt–Berman induction theorem
$\DeclareMathOperator{\Aut}{Aut}\DeclareMathOperator{\Ind}{Ind}$I believe I can prove the following induction theorem (modulo carefully checking a few details), and I would like to know whether this i …
- 13k
5
votes
A question on some computation of group cohomologies
This is not a complete answer, but it's too long for a comment. Here is what I would try (for $G=C_p\rtimes C_2$ for any $p$): your $M$ sits in the exact sequence
$$
0\rightarrow M\rightarrow \mathbb{ …
- 13k
6
votes
Accepted
Irreducible mod-p representation of a semidirect product with trivial p-core
The following is a direct proof that any extension of $G$ by $V$ splits. It is taken from a joint paper of mine with Tim Dokchitser, where the proof starts in the last paragraph of page 12.
First, no …
- 13k
35
votes
6
answers
5k
views
Character-free proof that Frobenius kernel is a normal subgroup?
The question is in the title, but here is some background/reminders:
A subgroup $H\neq\{1\}$ of a finite group $G$ is called a Frobenius complement if $H\cap H^g = \{1\}$ for all $g\in G\backslash H$ …
- 13k
13
votes
Branching rule from symmetric group $S_{2n}$ to hyperoctahedral group $H_n$
This is an answer to the second question: I ran an experiment with $S_6$ (which was the best guess due to the famous "oddness" of 6). There are two subgroups in $S_6$ isomorphic to this involution cen …
- 13k
7
votes
Accepted
Character table for the affine group of Z/p^nZ
The groups you are interested in are sometimes called false Tate extensions in number theorists' jargon. They are Galois groups of the Galois closures of extensions of $\mathbb{Q}$ obtained by adjoini …
- 13k