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2 votes
0 answers
102 views

Are the alternating pentultimate and the $2$-alternating pentultimate isomorphic?

Consider a physical puzzle which is in the shape of a dodecahedron where the pieces that move are the face centers and you can slice it through a plane parallel to two opposite faces going through the ...
Bram Cohen's user avatar
2 votes
0 answers
156 views

Special sets of involutions generating ${\rm S}_n$

For which positive integers $k$ and $r$ are there involutions $g_{n,i} \in {\rm S}_n$ $(n \in \mathbb{N}, \ i = 1, \dots, k)$ such that the following hold?: for any $n$, the $g_{n,i}$ $(i = 1, \dots, ...
Stefan Kohl's user avatar
  • 19.6k
1 vote
2 answers
207 views

Complexity of decision problem to decide if permutation group is $k$-transitive

Given a finite permutation group $G$ (a subgroup of the symmetric group on a finite set) in terms of its generators, what is known about the decision problem of deciding if $G$ is $k$-transitive for a ...
StefanH's user avatar
  • 798
1 vote
2 answers
1k views

Generators for permutation groups

Consider (e.g.) the full permutation group $G=S_6$. A valid set of generators and equations for $G$ is $r^6=m^2=(rm)^5=1$. I say this system has width $3$ (because there are $3$ equations), length $10$...
Hauke Reddmann's user avatar
1 vote
1 answer
80 views

Complexity to decide for permutation group if every element fixed at most $k$ points

I want to consider the following problem, which generalises the decision problem to decide if a given finite permutation group is a Frobenius group: Given a finite permutation group in terms of its ...
StefanH's user avatar
  • 798
1 vote
0 answers
90 views

What is an upper limit of relative size of conjugacy class of the transitive finite group?

What is $$ \limsup_{n\to\infty} \sup_{\deg(G)=n} \left( \frac{\max_{x\in G} \left|\operatorname{conj}(x)\right|}{\operatorname{ord}(G)}\right),$$ $G$ transitive permutation group? And what are the ...
Slepecky Mamut's user avatar
1 vote
0 answers
242 views

Counting elements having a given cycle structure in maximal subgroups of a generalized symmetric group

Let $G$ be the wreath product $C_7\wr S_{18}$, where $C_7$ is the cyclic group of order 7 and $S_{18}$ is the symmetric group on 18 symbols. Consider $G$ to be embedded in the group $S_{126}=S_{7\cdot ...
352506's user avatar
  • 1,021
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, $(...
Sebastien Palcoux's user avatar
1 vote
0 answers
221 views

A connection between nonplanar complete graphs and the alternating groups?

I didn't get any response on MSE so I though I'd give this a try here (my question on MSE). I went to an undergrad's senior honors thesis presentation a while ago. She was discussing crossing numbers ...
Bill Cook's user avatar
  • 1,197
0 votes
1 answer
213 views

A particular permutation on $\mathbb{Z}_n$

Let us consider $\mathbb{Z}_N=\{0,1,\cdots N-1\}$. Does there exist any permutation $\sigma$ on $\mathbb{Z}_N$ satisfying : $$\exists p,q\in\mathbb{N}, ~~p,q\neq0 (mod~N) ~~\forall i,j\in \mathbb{Z}...
ABB's user avatar
  • 4,058
0 votes
0 answers
142 views

Subgroups of powers of the alternating group on 5 elements

Definition: Let $G$ be a group, and let $H \leq G$ be a subgroup. We say that $H$ is big in $G$ if for every intermediate subgroup $H \leq L \leq G$ there exists some $x \in L$ such that $\langle H \...
Pablo's user avatar
  • 11.3k
0 votes
0 answers
229 views

Orbits of stabilizer of two points in a 2-transitive permutation group

I was doing something which needs to know sizes of all orbits of the stabilizer of two points in a 2-transitive permutation group. Since all 2-transitive permutation groups are known ans so are their ...
Binzhou Xia's user avatar

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