All Questions
Tagged with finite-groups symmetric-groups
68 questions
3
votes
1
answer
182
views
Schur cover of alternating groups
Wilson's book "The finite simple groups" gives (in section 2.7) a description of the double cover of the alternating groups. First, one constructs a double cover $2S_n$ of the symmetric ...
- 7,527
0
votes
1
answer
205
views
Hyperoctahedral group, preliminaries [closed]
I am looking for information on the hyperoctahedral group
From what I understand, the hyperoctahedral group is 'the generalized symmetric group' in the case where $m=2$. That is, the hyperoctahedral ...
- 111
5
votes
0
answers
200
views
Subgroups of the symmetric group and binary relations
Motivation
The following came up in my work recently. (NB this is the motivation, not the question I'm asking. You can skip to the actual question below, which is self-contained, but not self-...
- 756
2
votes
2
answers
210
views
is the embedding $\mathrm{Sp}_{2m}(p)\leqslant S_{p^m-1}$ possible?
Is the following embedding possible?
$\mathrm{Sp}_{2m}(p)\leqslant S_{p^m-1}$ where $S_{p^m-1}$ is a symmetric group and $p$ is prime. I see that when $p=3$ and $m=3$, the order of the former does ...
- 501
8
votes
0
answers
236
views
Alternate proof in Fulton–Harris of representation theoretic version of Littlewood–Richardson rule
$\DeclareMathOperator\Ind{Ind}$Let $d = d_1 + d_2$ with $d_1$, $d_2$ positive integers. Let $\lambda$ be a partition of $d_1$ and $\mu$ a partition of $d_2$, so that the Young symmetrizer construction ...
- 241
2
votes
0
answers
220
views
Characters of alternating groups
I am studying certain properties of characters of alternating groups and I have found precisely three characters not satisfying it up to $A_{15}$. These are:
A character of dimension $3.696$ of $A_{...
- 370
6
votes
1
answer
542
views
Is there a classification of homomorphisms $S_n \to S_{n+k}$ for small $k$?
Homomorphisms $B_n \to B_{2n}$ and $B_n \to S_{2n}$ have been classified in Chen–Kordek–Margalit - Homomorphisms between braid groups and Lin - Braids and permutations respectively. I am interested in ...
- 185
1
vote
0
answers
72
views
Scalars by which symmetrizations of cyclic permutations act on Specht modules
Let $S_n$ be the symmetric group. Pick $a \in 2,\ldots,n$ and denote by $c_a \in \mathbb{C}[S_n]$ the symmetrization of the element $(12\ldots a)$ i.e. $c_a$ is the sum of cycles of type $a$.
Let $\...
- 163
25
votes
6
answers
3k
views
What is the standard 2-generating set of the symmetric group good for?
I apologize for this question which is obviously not research-level. I've been teaching to master students the standard generating sets of the symmetric and alternating groups and I wasn't able to ...
- 4,492
4
votes
1
answer
214
views
A transitive action on a specific set
Let $G$ be a finite group and $\lambda\in G$, consider a set $$D^{p+*}_{G}(\lambda):=\{P\in S^{p+*}_G|P^{\lambda}=P=[P,\lambda]\}$$ where:
$S^{p+*}_{G}$ denotes the set consisting of all non-trivial $...
- 311
10
votes
7
answers
2k
views
Representations of products of symmetric groups
I'm writing a paper and want to cite some references to efficiently prove that over any field $k$ of characteristic zero, every irreducible representation of a product of symmetric groups, say
$$ S_{...
- 22.3k
7
votes
1
answer
344
views
For which $n$ can $S_n$ act transitively on $n+k$ elements?
It is known that the symmetric group $S_n$ can act transitively on $n+1$ elements if and only if $n=5$.
Are there similar classifications for $S_n$ acting transitively on $n+k$ elements, where $k$ is ...
- 13.6k
10
votes
2
answers
547
views
Arbitrarily large finite irreducible matrix groups in odd dimension?
I consider a finite irreducible matrix group $\Gamma\subseteq\mathrm{GL}(\Bbb R^d)$. I am interested in the maximal size of $\Gamma$ depending on $d$. But this question makes only sense if there is an ...
- 13.6k
1
vote
0
answers
213
views
Is there any research on the action of a subgroup on the whole finite group by conjugation?
I want to know whether there are any research on the orbits of the action of a subgroup by conjugation on the whole group, when the group is finite. (Especially whole symmetric group.)
I'm especially ...
- 1,013
2
votes
1
answer
656
views
Combinatorial problem in $\mathsf{S}_4$
I am working on a problem in Combinatorial Group Theory related to a construction in Algebraic Geometry, and I would like to have a conceptual proof of the fact described below.
I am looking for ...
- 66.3k
3
votes
0
answers
157
views
Faithful representation into $\operatorname{GL}(9,3)$
Take $T=\big(\left< (123) \right> \times \left< (456) \right> \times \left< (789) \right>\big) \rtimes \left< (147)(258)(369) \right> \leq S_9$.
Does there exist an injective ...
- 381
3
votes
1
answer
143
views
Permutation representation of a finite $p$-group
In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book," Theory of Groups Of Finite Order". The group ($\mathbb{Z_{p^{2}}}\rtimes \mathbb{Z_{p^{}}}) ...
- 381
3
votes
0
answers
400
views
Character table of the symmetric group $S_n$ according to James
In James, "The representation theory of the symmetric groups" an algorithm is described to produce the character table of a symmetric group. The proof involves the equation (pp. 22,23)
$$\...
- 2,050
4
votes
2
answers
485
views
Transposition Cayley graphs are planar
Consider the Cayley graph $G$ with vertex set the elements of the symmetric group $S_n$ and generating set the set of minimal transposition generators of the group $S_n$, that is the set $S=\{(12),(13)...
- 2,089
8
votes
0
answers
188
views
Non-zero group determinant for symmetric group
Let $G$ be a finite group. Given complex numbers $x=\{x_g: g\in G\}$, one can define a $|G|\times |G|$ matrix $X$, with entries $X_{g,h} = x_{gh^{-1}}$.
Let's consider $G$ being the symmetric group $...
- 1,839
14
votes
0
answers
262
views
Which irreducible representations of the symmetric group are eigenspaces of class sums?
In the setting of complex representations of finite groups, a class sum $1_C=\sum_{g\in C} g$ acts on an irreducible representation $V$ as $\lambda(C,V)\operatorname{Id}$, where $\lambda(C,V)=|C|\...
- 2,306
11
votes
2
answers
744
views
A criterion for finite abelian group to embed into a symmetric group
Let $G$ be a finite abelian group. Write $G\approx \mathbb{Z}/p_1^{i_1}\mathbb{Z}\times\dots \mathbb{Z}/p_m^{i_m}\mathbb{Z}$, with $m\ge 0$, $p_1,\dots,p_m$ primes (not necessarily distinct) and $i_k\...
user143954
3
votes
0
answers
133
views
Is there some sort of formula for $t(S_n)$?
Let $G$ be a finite group. Define $t(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > t(G)$ and $\langle X \rangle = G$, then $XXX = G$.
Is there some sort of formula for $t(S_n)...
- 2,618
9
votes
0
answers
114
views
Smith normal form of conjugacy class actions
This question was inspired by Smith Normal Form of a Cayley Graph of the Symmetric Group.
Let $\mathbb{Q}S_n$ denote the group algebra over $\mathbb{Q}$ of the
symmetric group $S_n$. Identify a ...
- 50.8k
6
votes
1
answer
262
views
Is there some sort of formula for $\tau(S_n)$?
Let $G$ be a finite group. Define $\tau(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > \tau(G)$, then $XXX = \langle X \rangle$.
Is there some sort of formula for $\tau(S_n)$, ...
- 2,618
11
votes
1
answer
550
views
Probability of words summing to $1$ in $S_n$ or $\mathrm{PGL}_2(n)$
$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Conj{Conj}$Let $G$ be the symmetric group $S_n$ or the projective general linear group $\PGL_2(n)$.
Let $X$ be a cyclically reduced word in the ...
- 9,501
3
votes
0
answers
57
views
Groups that can occur as graph automorphisms of a fixed size graph
From theorem $4$ and corollary $1$ in this book we have that graph isomorphism has to do with automorphism group of a graph. We also know every group is the automorphism group of a graph by Frucht's ...
- 13.9k
31
votes
2
answers
1k
views
Does the symmetric group $S_{10}$ factor as a knit product of symmetric subgroups $S_6$ and $S_7$?
By knit product (alias: Zappa-Szép product), I mean a product $AB$ of subgroups for which $A\cap B=1$. In particular, note that neither subgroup is required to be normal, thus making this a ...
- 1,068
10
votes
2
answers
1k
views
A cancellation property for permutations?
Let $S_n$ be the group of $n$-permutations. Denote the number of inversions of $\sigma\in S_n$ by $\ell(\sigma)$.
QUESTION. Assume $n>2$. Does this cancellation property hold true?
$$\sum_{\...
- 43.2k
4
votes
0
answers
266
views
Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)
In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...
- 24.9k
4
votes
1
answer
745
views
Decomposition into irreducible of a representation of the wreath product $S_d\wr S_n$
Let $S_d, S_n$ be the permutation groups of $d,n$ elements.
An intuitive representation of the wreath product $S_d\wr S_n$ is $V_1\otimes...\otimes V_n$, where each $V_i$ is of dimension $d$. Writing ...
- 583
5
votes
1
answer
365
views
Large subgroups of $S_n$ without large symmetric or alternating subgroups
I'm interested in determining the existence of a permutation group $G\subseteq S_n$ of the following form.
$G$ is large. Meaning that $G$ have at least $n!/2^{o(n)}$ elements. Equivalently, their ...
- 311
2
votes
0
answers
85
views
Permutation factorizations according to number of generated orbits
Let $\pi$ be a permutation in $S_n$ with cycle type $\lambda$.
How many factorizations into two factors $\pi=\sigma_1\sigma_2$ are there, such that the subgroup $\langle \sigma_1,\sigma_2\rangle$ ...
- 2,552
1
vote
2
answers
513
views
What are all the transitive extensions of cyclic groups?
"Let $G$ be a transitive group of permutations on a given set of letters. Let a new fixed letter be adjoined to every permutation of $G$. Then a transitive group $H$ of permutations on the ...
- 347
4
votes
1
answer
465
views
Is the Normal centralizer problem in P?
Notation
$\le$ is used for the subgroup relation;
$P$ means polynomial time in input size;
$\Omega = \{1,2,3,\cdots,n\}$ is a input domain;
$\mathrm{Sym}(\Omega)$ means the symmetric group on $\...
- 313
10
votes
3
answers
734
views
Low-dimensional irreducible 2-modular representations of the symmetric group
I apologize if this question is a little too basic for MathOverflow, but it's somewhat outside of my background and I'm frustrated that the answer doesn't seem to be explicit in the literature even ...
- 1,298
7
votes
2
answers
307
views
Generating symmetric groups with small cycles
This was asked but never answered at MSE.
Let $S_n$ denote the symmetric group and let $H$ be a subgroup which contains
an $n$-cycle. If $n$ is prime, and if $H$ also contains a 2-cycle, then ...
- 1,411
3
votes
0
answers
164
views
Generating sets of the symmetric group that yield isomorphic Cayley graphs
Let $S$ and $S'$ be subsets of size $k$ of $\mathfrak{S}_n$.
Are there any necessary or sufficient conditions to determine whether or not $S$ and $S'$ yield isomorphic Cayley graphs?
Assuming we ...
- 1,164
1
vote
0
answers
285
views
Classification of transitive subgroups of finite symmetric groups generated by double transpositions
I want to classify (up to isomorphism) all transitive subgroups of symmetric group $S_n$ which are generated by double transpositions (product of two transpositions). Is there a characterization for ...
- 11
16
votes
1
answer
484
views
Irreducible representations occuring in $\mathrm{Ind}_G^{S_{|G|}}1$ for $G$ finite group
Let $G$ be a finite group with $|G|=n$, let $S_G=S_n$ be the group of $n!$ permutations of the set $G$. Then $G$ is a subgroup of $S_G$ via left-translation (i.e. $g\in G$ corresponds to the ...
- 691
9
votes
2
answers
762
views
Solutions of $x^d=1$ in the symmetric group
L Moser and M Wyman, On solutions of $x^d = 1$ in symmetric groups, Canad. J. Math., 7 (1955), pages 159-168, explored asymptotic behavior of the cardinality of such permutations:
$$f_d(n):=\#\{\pi\in\...
- 43.2k
2
votes
1
answer
219
views
Symmetric group acting on the set of boolean functions
Let $S_n$ act on the set of boolean functions of size $n$ in the following way:
If $f$ is a boolean function and $\alpha \in S_n$, then $g=\alpha f$ and $g(x)=f(\alpha(x))$ where $x$ is boolean ...
- 337
3
votes
0
answers
226
views
$S_n$ action on the sequences of transpositions
It is well-known, that any element $\rho$ of the symmetric group $S_n$ with $n-p$ cycles admits a unique presentation as a product of a sequence of transpositions $\{(a_i\,b_i)\}_{i = 1}^p$ with $a_i &...
- 401
22
votes
1
answer
599
views
A symmetric-like group and the quaternion group $Q_8$
It is well known that the symmetric group $S_n$ admits presentation with
$\{(ij) \mid i\neq j\}$ as the set of generators and the following list of relations
(in every formula distinct letters denote ...
- 493
3
votes
4
answers
610
views
Factorization in the group algebra of symmetric groups
Let $S_n$ be the symmetric group on $\{1, \ldots, n\}$. Let
\begin{align}
T=\sum_{g\in S_n} g.
\end{align}
Are there some references about the factorization of $T$?
In the case of $n=3$, we have
\...
- 6,201
6
votes
1
answer
341
views
Sum of Young symmetrisers of a given shape
Preliminaries and notation:
Let $n\in \mathbb{Z}_{>0}$ and $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_s)\vdash n$ be a partition. Given a Young diagram of shape $\lambda$, we can associate it ...
- 87
6
votes
2
answers
532
views
A question about (unicity of certain cycles in a Cayley graph of a) symmetric group
Let $S=\{(1,2),(1,2,3,\ldots,n),(1,2,3,\ldots,n)^{-1}=(1,n\ldots,2)\}$ be a subset of the symmetric group $S_n$. We know that $(1,2,\ldots,n)(1,2)=(2,3,\ldots,n)$, and thus $$[(1,2,\ldots,n)(1,2)]^{n-...
- 377
5
votes
1
answer
204
views
A decomposition of $w_0$ which is similar to the reduced decomposition
Some basic definitions about reduced decomposition:
In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by adjacent ...
- 1,997
25
votes
3
answers
4k
views
Simplicity of alternating group $A_n$
I am teaching an introductory group theory course, and it has come to the inevitable proof that $A_n$ is simple for $n\geq 5$. Now, there seem to be a number of proofs that I can find – one the "...
- 96.4k
15
votes
2
answers
838
views
factorization of the regular representation of the symmetric group
Let $\mathbb{C}[S_n]$ be the regular representation of the symmetric group $S_n$, and let $\mathbb{C}^n$ be the vector representation.
Question: Does there exist a representation $V$ (of dimension $(...
- 2,537