The computational-group-theo tag has no wiki summary.

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### Mathematic Theory of Computation [on hold]

Let Sigma be an alphabet. Use the principle of induction on Sigma^*, to prove that |w v| = |w| + |v| for all w, v in Sigma^.

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### 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 ...

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### 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 ...

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### A dual version of a theorem of Øystein Ore in group theory

This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved:
Theorem: Let $(H \subset G)$ be an inclusion of finite groups such that the lattice ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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: ...

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### Does this class of groups contain finitely generated infinite periodic groups?

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
$\tau_{r_1(m_1),r_2(m_2)}$ be the ...

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### For which series of finite simple groups is it algorithmically decidable whether they contain a homomorphic image of a given finitely presented group?

Let $G$ be a group given by a finite presentation.
On the one hand, it is easy to determine the abelian invariants of $G$, or in other words,
it is algorithmically decidable whether $G$ surjects to a ...

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### Some resources about minimum-length generator sequences

In the group theory I want to know what are the best results known for problem of finding minimum-length generator sequences. This problam have different titles in articles that cause difficality in ...

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### 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
...

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### Using math software to show that the following groups are infinite?

I would like to show that the following finitely presented group in 3 generators $P, Q, R$ is infinite in certain cases:
$$P^p, Q^q, R^r, (PQ)^2, (QR)^2, (PQR)^2, (QR^{r/2+1})^a (RQR^{r/2})^b$$
For ...

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### 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 ...

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### Generator size for cyclic groups

Let $p$ be prime. Consider $\Bbb Z_{p}$, the cyclic multiplicative group.
Is it possible to choose a generator $c$ as small as $O(\log(p))$? (wiki shows $c$ as small as $O(\log^{6}(p))$ is possible ...

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### Computational Ring Theory

I have tried to understand and program CGT algorithms though I am a beginner still. But I never get to hear Computational Ring Theory. Even GAP largely supports Groups Theory. Is there some initiative ...

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### 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 ...

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### 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
...

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### Groups generated by 3 involutions

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
$\tau_{r_1(m_1),r_2(m_2)}$ be the ...

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### Characterization of the elements of an infinite simple group

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
$\tau_{r_1(m_1),r_2(m_2)}$ be the ...

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### Relations in a particular subgroup of the braid group.

I think this should be a 10 minute exercise in a decent computer algebra package - unfortunately I'm hopelessly ignorant of such things, so I'm putting it up here in the hope that someone will be kind ...

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### God's number for the $n \times n \times n$-cube

This is a question about Rubik's Cube and generalizations of this puzzle, such as Rubik's Revenge, Professor's cube or in general the $n \times n \times n$ cube.
Let $g(n)$ be the smallest number ...

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### Computability and complexity of computing $|Hom(G,H)|$ for finitely presented groups G, H.

In the general case, I want to say that determining $|Hom(G,H)|$ is incomputable, arguing that you could use the number to test for simplicity of a presentation, but I am new to this area and I keep ...

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### Necessary/Sufficient condition/Algorithm that tells me a function field is a kummer extension

I start my question with an example. Suppose $F/K$ be the function field generated by $x^n - yx^{n-1} - 1 = 0$. It is not a cyclic over K(y), but if I set $t = yx^{n-1}$ then we have $K(x,t) \subset ...

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### 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 ...

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### How to generate all finite groups of order n? [closed]

I know how to generate all Abelian groups of order n, but how would I generate the others? I can't seem to find anything about this.
By "generate", I mean produce the Cayley tables for all groups of ...

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### Computational Algebra - Where?

Hello.
I'm on my last semester of a math B.Sc. and about to start studying for a math M.Sc in the same institute.
It now seems like a good time to start thinking of a PhD.
I'm interested in both ...

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### Testing permutations to see if they generate $S_n$

Alright, so a similar question was recently asked about the theoretical bound for generating certain permutations in polynomial time. I had been thinking about a related problem in algorithms (with ...