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Results tagged with finite-groups
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user 22846
Questions on group theory which concern finite groups.
45
votes
Accepted
learning Deligne-Lusztig theory
You have given no indication as to your background, so the following imagines you don’t know anything. I have purposely left interesting things out as this is designed to get you from 0 to DL theory. …
- 2,902
31
votes
How do you *state* the Classification of finite simple groups?
I can't answer your general question but I can answer your side question. Almost all of the groups of Lie type are constructed as follows. You take a simple algebraic group $G$ defined over an algebra …
- 2,902
13
votes
Accepted
Analogy between product of conjugacy classes and irreps: is there analog of Thompson conject...
In the following article
Heide, Gerhard; Saxl, Jan; Tiep, Pham Huu; Zalesski, Alexandre E.
Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type. Proc. …
- 2,902
7
votes
Accepted
Unipotent orbit in adjoint group over finite field
That's not what they're claiming and your statement is not true. Your claim is that every unipotent element is rational. However Lemma 5.6 of this article by Tiep and Zalesskii provides a counter exam …
- 2,902
6
votes
Accepted
Principal series of finite group of Lie type
So, I think the answer to your question is yes. This may not be the slickest proof but I think it works. Firstly let $\mathrm{pr}_G$ be the projection map from the space of all class functions to the …
- 2,902
5
votes
Accepted
Finite field analogue of representations in same packet have equal central character
This is quite an old question but I believe the answer to your question is given in Lemma 2.2 of Malle's paper "Height 0 characters of finite groups of Lie type" (2007) which is freely available onlin …
- 2,902
5
votes
Accepted
Sylow $p$-subgroup of GL
Steinberg's "Lecture Notes on Chevalley Groups" the Corollary of Lemma 54 on page 132. There is possibly a more modern reference.
EDIT: Sorry the reference to Steinberg is not sufficient as he does n …
- 2,902
5
votes
Accepted
Centralizers of $\mathbb{F}_q$-rational semisimple elements of a finite group of Lie type
As @LSpice already pointed out, you need $q$ to be sufficiently large even in the case of a Levi subgroup. Just take $G = \operatorname{GL}_n(\overline{\mathbb{F}}_q)$ and $G^F = \operatorname{GL}_n(\ …
- 2,902
3
votes
Accepted
Is there any Lefschetz-like principle for representations of finite groups?
I was encouraged to make my comment an answer, so will do so.
If $G$ is a finite group and $\mathbb{K}$ is a field then many interesting results that can be proved using character theory can also be …
- 2,902
3
votes
Accepted
A bijection between Lusztig series induced by inflation
Your approach is correct and is proven in the book by Digne-Michel (in fact a more general statement is proven there). Indeed, by Proposition 13.22 in Digne-Michel we know that
$$R_{T\subseteq B}^G(\ …
- 2,902
1
vote
Regular elements in the torus of a group of Lie type
This is an old question now but I had cause to look at it recently. I thought it was worthwhile pointing out that Carter's proof about the existence of nondegenerate maximal tori in Proposition 3.6.6 …
- 2,902