All Questions
7 questions
4
votes
0
answers
115
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Complexity to find "short" (e.g. polynomial in diameter) decomposition of the permutation into the product of generators?
Question 1: Consider the symmetric group $S_n$ and some set of permutations $p_i$. Given permutation $g$ - what is known about the algorithmic complexity to decompose $g$ into product of $p_i$ ...
- 24.9k
10
votes
0
answers
194
views
Permutation groups with diameter $O(n \log n)$
I suspect that many permutation puzzles can be solved in $O(n \log n)$ moves, which has led me to the following question/conjecture:
Suppose that
1. $P_i$ for $i<k=O(1)$ are permutations on an $n$ ...
- 7,545
1
vote
0
answers
179
views
Are the finite groups inclusions, almost all relatively cyclic?
Definition: An inclusion of finite groups $(A \subset B)$ is relatively cyclic if $\exists b \in B$ such that $\langle A,b \rangle = B$.
Definition: Two inclusions of finite groups are equivalent, $(...
12
votes
0
answers
699
views
Solving a set of equations in a finite symmetric group
A standard way to find solutions to a finite set of equations in a finite symmetric group
${\rm S}_n$ is to take the equations as relators of a finitely presented group, to use
the low index subgroups ...
- 19.6k
2
votes
1
answer
254
views
Classification of indecomposable inclusions $(H \subset G)$ with $G$ decomposable
Definition: A group $G$ is indecomposable if: $G = G_1 \times G_2 \Rightarrow \exists i \ G_i = 1$.
We can generalize the notion of indecomposable from groups to inclusion of groups as ...
5
votes
0
answers
300
views
Uniqueness of the direct product decomposition of inclusions of finite groups
This post is a generalization of Uniqueness of the direct product decomposition of finite groups.
Here we look inclusions of finite groups $(H \subset G)$ instead of just finite groups.
Definition: ...
8
votes
2
answers
588
views
How hard is it to compute the diameter and the growth function of a finite permutation group of small degree?
Let $G \leq {\rm S}_n$ be a finite permutation group, and let
$S = \{g_1, \dots, g_k\}$ be a generating set for $G$ which is closed
under inversion and which does not contain the identity.
The growth ...
- 19.6k