All Questions
10 questions
28
votes
1
answer
2k
views
Multiplying all the elements in a group
Let $G = \{ g_i | i = 1, ...,n \}$ be a finite group and denote by $G!$ the multiset consisting of all the products of all different elements of $G$ in any order, that is
$$ G! = [ \prod_i g_{\sigma(i)...
- 943
1
vote
1
answer
154
views
The least common multiple of all degrees of a finite Coxeter group and indecomposable elements in the generalized cycle decomposition
This question is a follow-up of the previous question and especially the last comment therein.
Let $(W,S)$ be a finite Coxeter system with reflections $T$. Let $\ell_T$ be the reflection length. ...
- 1,066
10
votes
2
answers
1k
views
A cancellation property for permutations?
Let $S_n$ be the group of $n$-permutations. Denote the number of inversions of $\sigma\in S_n$ by $\ell(\sigma)$.
QUESTION. Assume $n>2$. Does this cancellation property hold true?
$$\sum_{\...
- 43.2k
4
votes
2
answers
144
views
Factorizations in terms of characters
I have asked this in math.SE (https://math.stackexchange.com/questions/2698772/factorizations-in-terms-of-characters) but it was barely viewed.
I have seen mention in different places that the number ...
- 1,549
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
5
votes
2
answers
245
views
Counting transitive generators according to coset type
Let $\sigma=(1\;2)(3\;4)\cdots (n-1\; n)$ be a fixed-point-free involution in $S_{2n}$. I want to count permutations $\pi$ such that the group $\langle \pi,\sigma\rangle$ generated by $\pi$ and $\...
- 1,549
2
votes
0
answers
110
views
How many symmetric strings a permutation fixes
Let $A$ be an alphabet of $N$ symbols. Let $S_n$ be the group of permutations of $n$ symbols. A permutation acts on a string of letters from $A$ in the obvious way.
If I ask, given a permutation $\pi\...
- 1,549
6
votes
0
answers
240
views
Factorization of permutations into two factors with fixed number of cycles, plus a placement condition
In my recent work I have been led to consider the following type of permutation factorizations.
Let $\pi_1$ and $\pi_2$ be two fixed permutations, with disjoint support, i.e. $\pi_1$ acts on $\{1,......
- 2,552
7
votes
2
answers
751
views
Looking for deterministic criteria to generate the symmetric group?
So let $S_N$ be the symmetric group of degree $N$. We think of it as a permutation group via its
natural action on the set $T=\{1,2,\ldots,N\}$.
Say that $H\leq S_N$ is a subgroup which acts ...
- 7,609
40
votes
1
answer
2k
views
Orders of products of permutations
Let $p$ be a prime, $n\gg p$ not divisible by $p$ (say, $n>2^{2^p}$). Are there two permutations $a, b$ of the set $\{1,...,n\}$ which together act transitively on $\{1,2,...,n\}$ and such that all ...
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