All Questions
Tagged with finite-groups symmetric-groups
68 questions
12
votes
1
answer
510
views
Can a large transitive permutation group need many generators?
let $G$ be a transitive permutation group acting on $\{1, \ldots, n\}$, and let $d(G)$ be the minimal number of generators of $G$. Is it true, that for $n\rightarrow\infty$ we have $\frac{d(G)\log|G|}{...
21
votes
1
answer
1k
views
Okounkov-Vershik approach to representation theory of $S_n$
This is a rather soft question. I was wondering if someone could explain on a fundamental and intuitive level, what the Okounkov-Vershik approach to representation theory of $S_n$ is all about. It's ...
- 291
16
votes
2
answers
722
views
Minimal maximal subgroup of the symmetric group
The question is pretty much in the title: What is the maximal subgroup of $S_d$ of maximal index (so minimal size)? A slight variant (I am not sure if it leads to a different answer) is: what if we ...
- 96.4k
8
votes
2
answers
464
views
Faithful projective representations of symmetric groups
This is a reference request.
Do you know where I can find the dimensions of the faithful projective representations of $S_n$ and $A_n$ for $n\ge 5$?
Thank you in advance.
- 845
14
votes
4
answers
2k
views
Number of squares in a finite group
This was asked at MSE but never answered.
Let $G$ be a finite group and denote by $sq(G)$ the number of squares in $G$ i.e. the number of elements in $G$ which possess a square root. For example, if
...
- 1,411
3
votes
2
answers
1k
views
The number of subgroups of ${\frak S}_n$
Because of my interest in this question, I listed the subgroups of ${\frak S}_n$ for $1\le n\le4$. I found that the number of subgroups are, respectively, $1,2,6,24$. It might be a coincidence, or it ...
- 52.3k
4
votes
1
answer
759
views
cohomology ring of symmetric group of order $3$
Let $S_3$ be the symmetric group of order $3$. What is the cohomology ring
$$
H^*(S_3;\mathbb{Z})?$$
My attempt: I want to use mathematical induction on $n$ for $S_n$.
For $n=1$, $S_1$ is trivial. ...
- 1,990
2
votes
0
answers
156
views
Special sets of involutions generating ${\rm S}_n$
For which positive integers $k$ and $r$ are there involutions $g_{n,i} \in {\rm S}_n$
$(n \in \mathbb{N}, \ i = 1, \dots, k)$ such that the following hold?:
for any $n$, the $g_{n,i}$ $(i = 1, \dots, ...
- 19.6k
8
votes
2
answers
1k
views
In general, are 'Young symmetrisers' given by Littlewood-Richardson 'Orthogonal projection Operators'?
Consider $V^{\otimes n}$ where $V$ is vector space and the representation of GL(V) acting in the usual way. Now if I consider tensor products or plethysms of irreducible spaces, this is not in general ...
- 349
1
vote
1
answer
226
views
Homomorphisms from irreducible spaces to reducible spaces
Let $P_{\lambda}$ be a Young symmetriser associated to the following tableau $(a_1 a_2 a_3 b_3 ; b_1 b_2)$ where the entries seperated by the ; belong to first and second COLUMNS of the tableau. Take $...
- 349
5
votes
1
answer
441
views
Minimum word length for an unusual set of generators of the symmetric group
Problem. Let $n\geq 2$ and let $T$ be the set of all permutations in $S_n$ of the form
$$t_k:=\prod_{1\leq i\leq k/2}(i,k-i) \qquad \hbox{for $k=3,4,\ldots,n+1$}.$$
Find the least integer $f_n$...
9
votes
2
answers
860
views
What is natural about the well-known bijection between conjugacy classes and irreps of a symmetric group?
Symmetric groups possess a well-known bijection between conjugacy classes and irreducible representations. More precisely, both sets are indexed by Young diagrams.
Question: To what extent is this ...
- 24.9k
11
votes
5
answers
2k
views
Structure of the adjoint representation of a (finite) group (Hopf algebra) ?
Every group acts on itself by conjugation $h \mapsto g h g^{-1}$. Respectively considering functions on a group we obtain a linear representation.
Question 1: what is known about this representation ...
- 24.9k
-3
votes
1
answer
267
views
A generalization of an old group problem [closed]
Here is an old exercise in group theory: (1) If $G$ is a group of order $2n$ with $n$ odd then $G$ is not simple and in fact $G$ has a normal subgroup of order $n$. I am going for one straight ...
user39125
8
votes
1
answer
400
views
Decomposition of $\mathrm{End}(V)$ as $S_n\times S_n$-module
Let $V$ be a finite-dimensional, complex vector space and set $\newcommand{\Gl}{\mathrm{Gl}}G:=\Gl(V)\times\Gl(V)$. Let $E:=\mathrm{End}(V)$ and consider its coordinate ring $\mathbb C[E]$, the space ...
- 4,902
5
votes
0
answers
418
views
What are the relation between Rep(G) and Rep(S_n)?
Let G be a finite group. We know it can be written as a subgroup of S_n. On the other hand, people sometimes say Rep(G) --- the category of all finite dimensional representations, are more interesting ...
- 2,040
29
votes
3
answers
4k
views
Roots of permutations
Consider the equation $x^2=x_0$ in the symmetric group $S_n$, where $x_0\in S_n$ is fixed. Is it true that for each integer $n\geq 0$, the maximal number of solutions (the number of square roots of $...
- 109k
73
votes
4
answers
4k
views
Is ${\rm S}_6$ the automorphism group of a group?
The automorphism group of the symmetric group $S_n$ is $S_n$ when $n$ is not $2$ or $6$, in which cases it is respectively $1$ and the semidirect product of $S_6$ with the (cyclic) group of order $2$. ...
- 1,069