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Results tagged with symmetric-groups
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user 28104
The symmetric group $S_n$ is the group of permutations of the set of integers $\{1,\dots,n\}$. This has $n!$ elements and is generated by the $n-1$ involutions exchanging consecutive integers. The symmetric groups form the simplest family of Coxeter groups.
6
votes
How many ways can a given permutation be obtained as a product of k 2-cycles?
For small enough $n$, an efficient way to perform this enumeration is described
in the solution to a GAP exercise I posed a few years ago.
It basically amounts to setting up a suitable matrix, raising …
- 19.6k
9
votes
How do most people write permutations?
The GAP convention is to multiply permutations from the left to the right, i.e.
$(1,2) \cdot (1,3) = (1,2,3)$, to write down each cycle with the smallest moved point first
and to sort cycles in ascend …
8
votes
Faithful projective representations of symmetric groups
See Schur's original paper
Schur, I. (1911), Über die Darstellung der symmetrischen und der
alternierenden Gruppe durch gebrochene lineare Substitutionen,
Crelle's Journal 139: 155–250. (EuDML)
- 19.6k
5
votes
Minimal maximal subgroup of the symmetric group
A table of the largest indices of maximal subgroups of ${\rm S}_d$ for small $d$
can be computed with GAP:
d | largest index of a maximal subgroup of S_d
------+--------------------------------- …
- 19.6k
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
Accepted
A question about (unicity of certain cycles in a Cayley graph of a) symmetric group
The smallest $n$ for which there exist sequences as asked for is $n = 7$:
$(1,2,3,4,5,6,7) \cdot (1,2) \cdot (1,7,6,5,4,3,2) \cdot (1,2)
\cdot (1,2,3,4,5,6,7) \cdot (1,2) \cdot$
$(1,7,6,5,4,3, …
- 19.6k