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8
votes
1answer
793 views

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

Let $(H \subset G)$ be an inclusion of finite groups. This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved: Theorem: $\mathcal{L}(H\subset G)$ ...
15
votes
0answers
643 views

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 ...
12
votes
0answers
422 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 ...
10
votes
0answers
328 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 ...
9
votes
0answers
182 views

Is an inclusion of finite groups with boolean lattice, linearly primitive?

Let $(H \subset G)$ be an inclusion 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 \ ...
5
votes
0answers
106 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: ...
5
votes
0answers
516 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 ...
4
votes
0answers
142 views

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 ...
4
votes
0answers
199 views

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 ...
3
votes
0answers
169 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: $G$ ...
2
votes
0answers
66 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 ...
2
votes
0answers
228 views

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 ...
1
vote
0answers
70 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 ...
1
vote
0answers
166 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
0answers
58 views

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 ...
1
vote
0answers
106 views

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 ...
0
votes
0answers
200 views

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 ...