21
votes
Does the symmetric group $S_{10}$ factor as a knit product of symmetric subgroups $S_6$ and $S_7$?
As has been pointed out in comments, the only subgroups $G$ and $H$ of $S_{10}$ isomorphic to $S_7$ and $S_6$ that could possibly have trivial intersections are the copy of $S_7$ that lies in $A_{10}$ ...
- 37.4k
21
votes
Accepted
Does the symmetric group $S_{10}$ factor as a knit product of symmetric subgroups $S_6$ and $S_7$?
Here are a few comments and a slightly different approach, though we take advantage of some of the earlier comments. We first note the well-known (at least to people who work with factorizations) ...
- 44.4k
17
votes
Accepted
Tensor power of the natural representation of Sn
This question was recently completely solved in this paper.
As explained on page 15 of the paper, letting $v_1,...,v_n$ be a basis for $V$, the standard basis vectors $v_{i_1} \otimes \cdots \otimes ...
- 2,743
17
votes
Accepted
What did Frobenius prove about $M_{12}$?
It seems to me that Frobenius is using lots of specific facts about the permutation group $M_{12}$ (and $M_{24}$, respectively) here, and that he has no doubts about the existence of these groups. In ...
- 6,919
17
votes
Accepted
Are there infinitely many insipid numbers?
Almost all $n$ are insipid. In fact, the number of non-insipid numbers at most $n$ grows like $2n/\log n$. See the paper
Cameron, Peter J.; Neumann, Peter M.; Teague, David N.
On the degrees of ...
- 3,291
16
votes
What did Frobenius prove about $M_{12}$?
This summary of Frobenius' 1904 paper might be of use:
Thomas Hawkins, The Mathematics of Frobenius in Context (page 527-528).
- 188k
16
votes
Accepted
Linear permutations commuting with $x\rightarrow x^{-1}$
The answer is no: a linear transformation of $F$ which commutes with $\phi$ is an automorphism of $F$.
This is a seemingly inelegant but simple argument.
Any map $\psi: F \rightarrow F$ can be ...
- 4,991
15
votes
Accepted
Is $(\mathbb{R},+)$ isomorphic to a subgroup of $S_\omega$?
See Theorem 4.3 of this paper by De Bruijn. Any abelian group of order $2^\kappa$ can be embedded in $Sym(\kappa)$ when $\kappa$ is infinite. (There is also an addendum to the paper which corrects ...
- 3,274
14
votes
Accepted
abelian quotients of permutation groups
In Finite permutation groups with large abelian quotients Kovacs and Praeger show (among other things) the following: If $G$ is a (not necessarily transitive) permutation group of degree $n$ and if ...
- 22.5k
14
votes
Accepted
Does every transitive permutation group contain a permutation whose cycle lengths have a common divisor?
There is a theorem of Fein, Kantor and Schacher that every finite transitive permutation group of degree at least two contains a derangement of prime power order, and hence all the cycle lengths are ...
- 2,168
12
votes
Accepted
Order of products of elements in symmetric groups
The main theorem in a paper of G. A. Miller [1] is the following:
THEOREM. If $l, m, n$ are any three integers greater than unity, of which we
call the greatest $k$, it is always possible to find ...
- 2,217
12
votes
Number of permutations that are products of disjoint cycles of distinct length
If we denote by $c_i(\sigma)$ the number of cycles of length $i$ in $\sigma$, we can write the exponential generating function of permutations with cycle statistics as
$$\sum_{n\geq 1}\sum_{\sigma\in ...
- 85.6k
11
votes
How do I know if an irreducible representation is a permutation representation?
The map from the Burnside ring to the representation ring is well studied. Andreas Dress wrote many papers on this in the 1970's. See also work of Tammo tom Dieck, who was applying things to ...
- 11.1k
10
votes
Accepted
sum-sets in a finite field
It took me some effort to find the references you were requesting in your comment, but here they are eventually:
Ordering subsets of the cyclic group to give distinct partial sums (posted to MO ...
- 23k
10
votes
Accepted
Time Complexity of the Word Problem for Finite Permutation Groups
As has been pointed out in comments, you cannot hope to do better in general than $O(n(l_1+l_2))$, where $n$ is the degree of the permutation groups and $l_1$, $l_2$ are the lengths of the words.
But ...
- 37.4k
10
votes
Questions about algorithms for permutation groups
The following is only an answer to the first question: Consider the subgroups $G_1=\langle (12)(34),(13)(24)\rangle$ and $G_2=\langle (12)(34),(34)(56)\rangle$ of $\Sigma_6$ which are both isomorphic ...
- 1,907
10
votes
Accepted
Is this known? As $p,q\to\infty$, most elements of the power set of $\{1,\dots,p\}\times\{1,\dots,q\}$ are in free $\Sigma_p\times\Sigma_q$-orbits
We'll show that in the regime $q\ge p>(2+\delta)\log_2q$, $p,q\to\infty$, the portion of subsets of $A\times B$ with $|A|=p$, $|B|=q$ that have non-trivial automorphisms is at most
$$
O(2^{-p}p^2q^...
- 61.9k
9
votes
Automorphism group of a special commuting graph
It's an exercise to check it coincides with $\mathrm{Aut}(S_6)$ which has $S_6$ as subgroup of index 2.
Here are the steps. First, for arbitrary $n\ge 6$, consider the graph of transpositions. So ...
- 63.9k
9
votes
Accepted
How many steps are required for double transitivity?
It seems that this is a lower bound of $\Omega(n^2)$.
Take an $n$ and an $a=\Theta( n) $ coprime with $n$ (with $a<n/2$). Then the permutations $\sigma=(12\dots n) $ and $\tau=(1, a+1) $ generate $...
- 23.7k
9
votes
For which $n$ can $S_n$ act transitively on $n+k$ elements?
Fix $k > 0$.
Suppose that $n > 6$ and $\frac{n(n-3)}{2} > k$. If $[S_n : H] \leq n+k$, then $H$ is one of the following: $S_n$, $A_n$, $S_{n-1}$, or $A_{n-1}$. So in particular if $[S_n : H] =...
- 2,217
9
votes
Accepted
Is there a known classification of regular multiplicity-free permutation groups?
These are the abelian regular permutation groups. The permutation character in this case is the character of the regular representation and in the regular representation a character appears with ...
- 38.6k
9
votes
Does the Okounkov-Vershik approach to the representation theory of $S_n$ shed new light on the problem of computing Kronecker coefficients?
Well the Okounkov-Vershik approach is close to 20 years old now, and it has been fairly well adopted by the community at large. So I won't say it could never help with the Kronecker coefficient ...
- 2,242
8
votes
What did Frobenius prove about $M_{12}$?
Not really an answer, and probably contained in Frobenius's paper, but a starting point might be that (using a result now usually credited to Blichfeldt) if $G$ is a sharply $5$-transitive group (of ...
- 44.4k
8
votes
Accepted
sum of character product over derangements
Using standard symmetric function notation, we have
\begin{eqnarray*} \sum_{n\geq 0}\sum_{\lambda,\mu\vdash n}
\frac{1}{n!}\left(\sum_{\pi\in D_n}\chi_\lambda(\pi)\chi_\mu(\pi)\right)
s_\...
- 50.8k
8
votes
Accepted
A sum over partitions involving "subpartitions"
Consider the generating functions
$$
R_k(u) = \sum_{j_1,\ldots,j_k\geq 0} u^{j_1 + 2 j_2 + \cdots + k j_k} \prod_{t=1}^k \frac 1{j_t! t^{j_t}} f_t(j_1,\ldots,j_t).
$$
I will prove by induction on $k$ ...
- 5,186
8
votes
Accepted
If $n=2m$, what is the order of the permutation $\sigma(k)=2k , \quad \sigma(m+k)=2k-1$
By adding a fixed point at $0$ (which preserves the order), the permutation $\sigma$ considered is just the multiplication by $2$ modulo $2m+1$. Thus, for $k \ge 0$, $\sigma^k$ is the identity map if ...
- 3,808
7
votes
Accepted
simply primitive permutation groups of degree $2p^2$
For the sake of getting this off the unanswered stack... Yes, all such actions are known.
By studying O'Nan--Scott--Aschbacher, one sees immediately that a group $S$ that has a primitive action of ...
- 11.2k
7
votes
Accepted
Permutation groups having a regular cyclic subgroup and a conjectured algebra of characters
The conjecture is true and holds, in a generalized version, whenever $K$ is an abelian regular subgroup. In fact by Theorem 1.9(d) in O. Tamaschke, Zur Theorie der Permutationsgruppen mit regulärer ...
- 11.2k
7
votes
Accepted
Group action with unique word
Fix $c\in [n]$. Let $\mathcal R_m(c)$ be $\{f_w(c)\colon |w|=m\}$ and $r_m(c)=|\mathcal R_m(c)|$. We define $\mathcal R_0(c)$ to be $\{c\}$.
Claim: Let $m\ge 0$. If $r_m(c)=r_{m+1}(c)$, then for all $...
- 23.2k
7
votes
Irreducible deleted permutation module for a finite group
As Padraig Ó Catháin explained in a comment, in coprime characteristic, this module is irreducible whenever $G$ is doubly transitive, and this is a sufficient condition when the field $k$ is ...
- 37.4k
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