Questions tagged [finite-groups]
Questions on group theory which concern finite groups.
117 questions from the last 365 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
3
votes
0
answers
192
views
permutations of finite groups
I am probably missing something here, but this looks like a question that should be easy.
Let $G$ be a finite group. The general question is,
for which groups does there exist a permutation, $\sigma$, ...
- 4,829
8
votes
1
answer
423
views
Groups with a certain character-theoretic property
Let $G$ be a finite group and let $\operatorname{Irr}(G)$ denote the set of all inequivalent, irreducible characters of $G$. Suppose $G$ be a group such that $\chi(g)+\chi(g^{-1})\in\mathbb{Z}$ for ...
- 203
0
votes
0
answers
114
views
Results on the possible orders for composition series factors for a finite group?
I have a somewhat general question about how much is known in the literature about the possible orderings of the composition series factors for a finite group. I am aware that for finite nilpotent ...
- 479
18
votes
1
answer
656
views
Prove that except for $n=1$, there are more topologies of size $n$ than groups of order $n$
This question was first asked here, but no answer was given.
For every positive integer $n$ let $a_n$ be the number of groups of order $n$ up to isomorphism, and let $b_n$ be the number of topological ...
- 81
7
votes
1
answer
284
views
Splitting of $\mathrm{GL}(n,\mathbb{Z}/4\mathbb{Z})$
For which $n$ does $\mathrm{GL}(n,\mathbb{Z}/4\mathbb{Z})$ split over its elementary Abelian normal subgroup of order $2^{n^2}$ with quotient $\mathrm{GL}(n,\mathbb{F}_2)$? The answer is trivially yes ...
- 3,461
3
votes
1
answer
140
views
Partition of a $2n$-dimensional vector space into non-parallel $n$-dimensional affine subspaces
Problem. Let $\mathcal A$ be a family of pairwise disjoint $n$-dimensional affine subspaces covering a $2n$-dimensional vector space over a finite field. Are any affine subspaces $A,B\in\mathcal A$ ...
- 46k
0
votes
1
answer
61
views
Can the alternating group be generated by the sharply 2-transitive union of cosets of sharply transitive Boolean subgroups
Question. Can the alternating group $\mathrm{Alt}(X)$ on a finite set $X$ of cardinality $|X|=n$ be generated by a sharply $2$-transitive set $\bigcup_{i=1}^{n-1}B_if_i$ such that for every $i<n$, $...
- 46k
2
votes
0
answers
202
views
Prove all irreps of the symmetric group are real
I have been trying to prove that every irrep $V$ of the symmetric group is real, that is, there exists a $S_n$-invariant symmetric nondegenerate bilinear form.
Since every permutation is conjugate to ...
- 21