All Questions
36 questions
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 ...
- 5,654
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
13
votes
2
answers
2k
views
Generalization of a theorem of Øystein Ore in group theory
Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive.
Proof: see below.
Let $(H \subset G)$ be an inclusion of finite groups and $\...
- 2,183
6
votes
1
answer
2k
views
Generalization of the fundamental theorem of cyclic groups
Let $G$ be a finite group then the fundamental theorem of cyclic groups can be formulated as follows:
Theorem: $G$ is cyclic iff it admits no two different subgroups with the same order.
proof: see ...
- 2,821
4
votes
4
answers
485
views
What are the rank 3 boolean intervals [H,G], with G simple group?
The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$.
The lattice $B_{3}$ is the following:
Question: What are the rank $3$ boolean intervals of the form $[H,G]$, with $...
- 2,773
15
votes
1
answer
821
views
Conjugated subgroups in $\mathsf{GL}(m+n,\mathbb{Z})$ implies conjugated subgroups in $\mathsf{GL}(n,\mathbb{Z})$?
In my research I came up with the following question:
Question: Let $H_1$ and $H_2$ be finite abelian subgroups of $\mathsf{GL}(n,\mathbb{Z})$. Define $$ H_1'=\left\{\begin{pmatrix} I_m &0\\0&...
- 156k
0
votes
0
answers
274
views
Algorithm to compute automorphism group of a finite group
Is there an algorithm to compute automorphism group of a finite group?
GAP has a function to do this, but while perusing their GitHub repo, I could not find an implementation. I'm struggling to find ...
- 63.9k
1
vote
2
answers
411
views
Are the character degrees determined by the conjugacy class sizes?
The computation below (part 1) shows that if two finite groups of order at most $100$ have the same (ordered) list of conjugacy class sizes, then they also have the same (ordered) list of (irreducible)...
9
votes
2
answers
811
views
Groups without factorization
A group G is said to have a factorization if there exist proper subgroups $A$ and $B$ such that $G = AB = \{ ab \ | \ a \in A, b \in B \}$.
The paper Factorisations of sporadic simple groups (...
- 44.4k
1
vote
1
answer
431
views
The sporadic numbers
Let call $n$ a sporadic number if the set of groups $G \neq A_n,S_n$ having a core-free maximal subgroup of index $n$ is non-empty and contains only sporadic simple groups.
By GAP, the set of all the ...
- 37.4k
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,...
- 44.4k
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 ...
1
vote
1
answer
318
views
How quickly can one compute the Hurwitz action of braid groups on finite groups?
Let $G$ be a finite group. Define the Hurwitz action of $B_{n}$ on $G^{n}$ by letting
$(x_{1},...,x_{n})\sigma_{i}=(x_{1},...,x_{i}x_{i+1}x_{i}^{-1},x_{i},x_{i+2},...,x_{n})$. I wonder what algorithms ...
8
votes
2
answers
586
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 ...
- 11.2k
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
3
votes
0
answers
164
views
Generating sets of the symmetric group that yield isomorphic Cayley graphs
Let $S$ and $S'$ be subsets of size $k$ of $\mathfrak{S}_n$.
Are there any necessary or sufficient conditions to determine whether or not $S$ and $S'$ yield isomorphic Cayley graphs?
Assuming we ...
- 63.9k
7
votes
1
answer
565
views
Are the distributive permutation groups linearly primitive?
An action of a group $G$ on a set $X \neq \emptyset$ is called transitive if $\forall x,y \in X$, $\exists g \in G$ such that $g.x = y$.
It is called primitive if it is transitive and preserves no non-...
9
votes
1
answer
235
views
Is a boolean interval of finite groups linearly primitive?
Let $[H,G]$ be an interval of finite groups.
Definition: Let $W$ be a representation of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{H}:=\{w \in W \ \vert \ kw=w \ , \forall h \...
3
votes
0
answers
128
views
Extension of Tits' theorem on groups with a BN-pair of rank ≥ 3
Tits has proved that a finite simple group $G$ with a BN-pair of rank $n \ge 3$, is of Lie type. Let $B$ be the Borel subgroup and $(W,S)$ the Coxeter system. The subset lattice of the set $S$ is ...
0
votes
0
answers
111
views
minimal permutation representations [duplicate]
Suppose I have a finite group $G.$ How hard is it to find the (a?) minimal degree permutation representation of $G?$ The second part of the question is: is there a table of such (hopefully for ...
- 63.9k
5
votes
0
answers
95
views
Is there an interval of finite groups, at index n, with strictly more elements than the subgroup lattice of any group, of order n?
Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice.
Let $s(n):= max\{|\mathcal{L}(G)| \text{ for } |G|=n \}$.
There is an OEIS page for the sequence $s(n)$: A018216
1, 2, 2, 5, 2, ...
CommunityBot
- 1
2
votes
2
answers
365
views
A good upper-bound for the cardinal of an interval of finite groups
This post is a relative version of General bound for the number of subgroups of a finite group
Let $[H,G]$ be a interval of finite groups with $|G:H| = n$.
Question: What is a good upper-bound of $|[...
CommunityBot
- 1
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 ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
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 \...
CommunityBot
- 1
0
votes
1
answer
81
views
About $c(A)$ in $c(A)|A|\leq |A^{-1}A|$
Let $G$ be a finite group, $\emptyset\neq A\subseteq G$, $A^{-1}:=\{ a^{-1}:a\in A\}$, and put $$c(A):=\max\{t\in \mathbb{Z}: t|A|\leq |A^{-1}A|\}$$
It is clear that $1\leq c(A)\leq \frac{|G|}{|A|}$, ...
- 1,281
3
votes
1
answer
499
views
Is there a way to find an efficient set of relations for presenting the subgroup generated by two matrices in $SL(2, q)$?
Given two elements $a, b \in SL(2, \mathbb{F}_q)$, is there a way to find an efficient presentation $$\langle x, y \mid \text{relations}\rangle$$ of the subgroup $\langle a, b \rangle$?
My intention ...
- 37.4k
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, $(...
1
vote
1
answer
225
views
Is a prime index inclusion of finite groups, separating?
Let $(H \subset G)$ be an inclusion of finite groups.
Let $\{ g_i \ \vert \ i \in I=[1, \dots ,n] \}$ a subset of $G$ of double coset representatives, i.e. $$G = \coprod_{i \in I} Hg_iH$$
On the ...
CommunityBot
- 1
4
votes
0
answers
199
views
Generalization of the fundamental theorem of cyclic groups 2
This post is a sequel of Generalization of the fundamental theorem of cyclic groups
Let $G$ be a finite group then the fundamental theorem of cyclic groups can be formulated as follows:
Theorem: $G$ ...
CommunityBot
- 1
5
votes
1
answer
326
views
Is there a nonabelian finite simple group with Grothendieck ring of multiplicity one?
Let $G$ be a finite group. It admits finitely many irreducible complex representations $H_1, \dots, H_r$ which generate, for $\oplus$ and $\otimes$, the Grothendieck ring $\mathcal{G}(G)$ of $G$ (also ...
- 44.4k
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 ...
- 118k
5
votes
1
answer
165
views
Can any finite distributive weighted lattice be realized by inclusion of groups?
By theorem 2.1 here, any finite distributive lattice $\mathcal{L}$ can be realized as an intermediate subgroups lattice.
A weighted lattice $(\mathcal{L},\tau)$ is a lattice $\mathcal{L}$ with a ...
- 11.2k
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: ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
4
votes
1
answer
446
views
Finding groups of odd order without non-cyclic nilpotent quotients
I hope that my question is appropriate for MO, since it might turn out te be mainly a question about GAP or other group theory software.
Is there an algorithm to produce all non-nilpotent groups of ...
- 5,654