All Questions
Tagged with co.combinatorics computational-group-theory
13 questions
14
votes
2
answers
1k
views
The Simultaneous Conjugacy Problem in the symmetric group $S_N$
We are interested in the following notions in the case $G=S_N$, the symmetric group on
$\{1,\dots,N\}$.
Fix a group $G$ and a number $d$. For $(g_1,\dots,g_d)\in G^d$ and $x\in G$, define
$$(g_1,\...
- 3,104
12
votes
0
answers
558
views
Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers
Definition / Question
Definition: Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where
$0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition
$...
- 19.6k
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
9
votes
1
answer
650
views
A stronger version of a problem of Kenneth Brown using representations
Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice. Let $\mu$ be the Möbius function on $\mathcal{L}(G)$.
The reduced Euler characteristic of the order complex of the coset poset $\{ ...
8
votes
1
answer
1k
views
GAP cannot solve Rubik's cube 4x4x4 and higher ? (Practical limits of Schreier–Sims algorithm)
According to our practical experiments and literature search - computer algebra system GAP cannot "solve" Rubik's cube 4x4x4 and higher. That means cannot decompose given random element of ...
- 24.9k
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
8
votes
0
answers
435
views
A relation between intersection and product on Boolean interval of finite groups
Let $[H,G]$ be a Boolean interval of finite groups (i.e. the lattice of intermediate subgroups $H \subseteq K \subseteq G$, is Boolean). For any element $K \in [H,G]$, let $K^{\complement}$ be its ...
7
votes
0
answers
1k
views
Example of a group with unsolvable word problem
Today I noticed that the last relator in the 27-relator presentation
of a group with unsolvable word problem given in
Donald J. Collins: A simple presentation of a group with unsolvable word problem.
...
- 19.6k
6
votes
1
answer
629
views
Positivity of the alternating sum of indices for boolean interval of finite groups
Let $G$ be a finite group and $H$ a subgroup such that the interval $[H,G]$ is a boolean lattice.
Let $L_1, \dots , L_n$ be the maximal subgroups of $G$ containing $H$.
Let the alternative sum ...
4
votes
0
answers
115
views
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
3
votes
1
answer
193
views
Maximal factorization of finite simple groups and no extra intermediate
The book The maximal factorizations of the finite simple groups and their automorphism groups (by Martin W. Liebeck, Cheryl E. Praeger and Jan Saxl) provides a classification of all the triples $(G,A,...
2
votes
0
answers
154
views
Nonvanishing of the dual Euler totient on boolean intervals of finite groups
The rank $n$ boolean lattice $B_n$, is the subset lattice of $\{1,2, \dotsm n \}$.
Let $[H,G]$ be a boolean interval of finite groups. Its Euler totient is defined by $$\varphi(H,G):=\sum_{K \in ...
1
vote
0
answers
81
views
An optimal lower bound related to generators in a boolean interval of finite groups
Let $[H,G]$ be a rank $n$ boolean interval of finite groups (i.e. $[H,G] \simeq B_n$ as lattice).
Let the set $E = \{ g \in G \ | \ \langle H,g \rangle = G \}$
Remark: If $g \in E$ then $Hg \...