Questions tagged [finite-groups]
Questions on group theory which concern finite groups.
7 questions from the last 30 days
6
votes
1
answer
111
views
The principal block of a Frobenius group
Let $G$ be a finite group. Fix a prime $p$. Let $P$ be a Sylow $p$-subgroup such that $P\cap P^x=1$ for all $x\not\in P$. (In other words, $P$ is a Frobenius complement.)
It follows from Frobenius' ...
- 2,603
5
votes
0
answers
103
views
A query about factorizable simple groups
Let $G$ be a finite simple group and $M_1,M_2,M_3,M_4$ be four distinct non-conjugate maximal subgroups of $G$ such that $M_1M_2=G=M_3M_4$. I feel the following is true:
There exist $i\in \{1,2\}$ and ...
- 638
1
vote
0
answers
49
views
Closest standard framework for a cyclically ordered 8-set with a fixed-point-free involution?
I am studying the following finite structure.
Let
$$
R=\{2,3,4,5,6,7,8,9\}, \qquad
L=(2\,3\,4\,6\,5\,8\,7\,9), \qquad
\delta=L^4=(2\,5)(3\,8)(4\,7)(6\,9).
$$
So $R$ is equipped with a cyclic order (...
4
votes
1
answer
224
views
Decomposing finite groups into unions of transversal normal subgroups
Problem. Assume that a finite group $G$ is the union $G=\bigcup_{i=1}^nH_i$ of $n\ge 2$ nontrivial normal subgroups $H_i$ such that $H_i\cap H_j=\{e\}$ for all distinct indices $i,j\le n$. Is $G$ ...
- 46k
1
vote
0
answers
135
views
+50
Spectrally undetermined finite orthogonal groups
For a representation $\rho$ of a group $G$, we denote by $\operatorname{tr}(\rho)$, the multiset $\{\{\operatorname{tr}(\rho(g)):g\in G\}\}$. Similarly, we denote by $\operatorname{sp}(\rho)$ the ...
- 369
1
vote
1
answer
144
views
Intersection of centralizers in $S_n$
Let $Z(g_1)$ and $Z(g_2)$ be the centralizers of two permutations $g_1$ and $g_2$ in the symmetric group $S_n$. Is there an algorithm which calculates the intersection $Z(g_1) \cap Z(g_2)$ as a ...
6
votes
1
answer
270
views
Can we index Deligne–Lusztig series by rational conjugacy classes?
Setting: Let $G$ be a reductive group over $\mathbb{F}_q$, such as $\text{SL}_n$. (The question will only be nontrivial when $Z(G)$ is disconnected.)
Background: In the study of the representation ...
- 1,139