# Tagged Questions

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### Finite groups generated by 3 involutions interchanging disjoint residue classes of the integers

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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### Torsion-free, normal subgroups of certain Coxeter groups

Let $G$ be the reflection group of a regular, 4-dimensional, hyperbolic honeycomb. I would like to find a family $H_i < G$ of finite-index, torsion-free subgroups of $G$, so that I can represent ...
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### 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 $1\leq c(A)\leq \frac{|G|}{|A|}$,...
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### programming to compute kernel quotient image of a $\mathbb{Z}$-module endomorphism

Let the integers $n\geq 2$, $k\geq 1$, $v=0$ or $1$ and $n_1,\cdots,n_k\geq 1$ such that $$\sum_{i=1}^k n_i+v=n.$$ Define $P_a^b=0$ if both $a,b$ are odd and $P_a^b={{[a/2]}\choose {[(a+b)/2]}}$ ...
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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 non-...
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### 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 ...
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### Computing a transversal of a subgroup $H$ of $G$ in expected $O(|G : H|^2 \log |G : H| + |H|)$ time

I have the book "Handbook of Computational Group Theory", by Derek Holt, and in it is a section on finding the transversal of a subgroup. Recall a transversal of a subgroup $H$ of $G$ is a single ...
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### 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: $G$ ...
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### 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 ...
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### Is there a way of canonically labelling permutation groups?

When working with large numbers of graphs, a canonical labelling routine is essential as, after the initial cost of canonically labelling each graph, it permits isomorphism checks to be replaced with ...
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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $m$,...
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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 ...
### 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 ...