The permutation-groups tag has no wiki summary.
2
votes
0answers
75 views
A section from subfactors to transitive groups
A finite group-subgroup subfactor is a subfactor $(N \subset M)$ isomorphic to $(R^G \subset R^H)$ with $(H \subset G)$ an inclusion of finite groups acting as outer automorphism on the hyperfinite ...
3
votes
1answer
142 views
Permutation groups transitive on partitions into ordered pairs
The following came up in a problem on graph reconstruction. It isn't very important, but I thought some people here might find it interesting and not too trivial (I'm not a group theorist).
Take a ...
3
votes
0answers
117 views
Largest permutation groups without “non-mixing” subgroups
We say that a subgroup of ${\rm Sym}(\mathbb{N})$ has sparse orbit representatives
if it has infinitely many orbits on $\mathbb{N}$, but the set of smallest orbit
representatives has natural density 0 ...
13
votes
1answer
264 views
Does O'Nan-Scott depend on CFSG?
My question is in the title. Some context: there are two versions of the O'Nan-Scott theorem. The first, weaker version, is due to O'Nan and Scott (independently) and gives the structure of the ...
6
votes
0answers
150 views
Permutation Group Question
A question about permutation groups: I wonder if someone
who is expert in permutation group theory could answer the
following question.
Let $x \in S_n$ (the symmetric group) be an involution which
...
2
votes
1answer
161 views
Doubly primitive groups with simple socle
The classification of doubly transitive groups with simple socle is
known. A good account of such classification can be found for example
in this paper:
Cameron, Peter J. Finite permutation groups ...
5
votes
1answer
154 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 ...
2
votes
5answers
239 views
Equivalence relations not associated with a group
This is a vague question; so vague that I wonder if anyone will get it. Many, perhaps most, equivalence relations that are regularly used in mathematics correspond to the orbits of some group action ...
4
votes
2answers
201 views
Palfy's theorem for nilpotent groups?
P. P. Palfy proved that a primitive solvable subgroup of $S_n$ has order bounded by $24^{-1/3} n^{3.24399\dots}$ (in: Pálfy, P. P.
A polynomial bound for the orders of primitive solvable groups.
J. ...
4
votes
0answers
166 views
Permutations with all cycles odd
I have recently needed to compute the number of permutations in $S_n$ with all cycles odd, and while this is easy using Flajolet-Sedgewick theory (the exponential generating function seems to be ...
1
vote
1answer
171 views
Group with 2 orbits on the nonnegative integers — description of the orbits
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 ...
4
votes
2answers
404 views
Transitive subgroup of $S_p$ containing a $p$-cycle and a double transposition
Let $p$ be a prime other than 5 or 7. Are $A_p$ and $S_p$ the only subgroups of $S_p$ that contains a $p$-cycle and a double transposition?
As for $p = 5$, the dihedral group $D_{10}$ contains a ...
0
votes
1answer
51 views
Transformation terminology question
Given a transformation $t$ from the transformation semigroup $T_{n}$, if you take powers of $t$ under composition you get a length $s$ stem followed by a cycle. Permutations by definition have a ...
4
votes
0answers
182 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
...
8
votes
1answer
178 views
Permutations of prescribed cycle types that multiply to the identity
Suppose that $\lambda_1,\lambda_2,\lambda_3$ are partititions of $n$. When do there exist permutations $\sigma_1,\sigma_2,\sigma_3 \in S_n$ such that
(1) $\sigma_1\sigma_2\sigma_3$ is the identity;
...
0
votes
2answers
122 views
the symmetric group $S_{2^{r−1}}$
Is there any routine technique to find a set of permutations which generate a Sylow 2-subgroup of the symmetric group $S_{2^{r−1}}$?
4
votes
3answers
329 views
automorphisms of graphs and finite permutation groups
I am interested in automorphisms of graphs and in using tools from permutation groups (especially such as in Wielandt's text on finite permutation groups, which I have been studying). What are some ...
5
votes
0answers
96 views
Information about permutation character from local action
Let $G$ be a finite permutation group acting transitively, but not regularly, on a set $V$. Let $H$ be the stabilizer of some point $v\in V$, and suppose that $H$ acts 2-transitively on one of its ...
7
votes
1answer
272 views
Permutation character of the symmetric group on subsets of certain size
The symmetric group $S_n$ acts on $[n]:=\{1,\ldots,n\}$, thereby inducing an action on the set $$\wp_k(n)=\{\: A\subseteq[n] \::\: \#A=k \:\}$$ of subsets of cardinality $k$, simply by $$(g,A)\mapsto ...
2
votes
1answer
185 views
Classification of generously transitive groups
A permutation group $G \lt S_n$ is called generously transitive, if for each $i,j$ there exists a permutation that interchanges them. Is there a reasonable classification of such (finite) groups?
1
vote
2answers
143 views
Is there any relation between automorphism group of a Cayley graph over a group and over its subgroup?
Let $\Gamma=Cay(G,S)$ be a Cayley graph over a group $G$, $H$ be a proper subgroup of $G$ and $\Sigma=Cay(H,T)$ where $S$ and $T$ are inversed-closed subsets of $G$ and $H$ not containing idendity, ...
4
votes
2answers
149 views
Maximal order of a metacyclic transitive permutation group of degree $n$
What is the maximal order $f(n)$ of a metacyclic (metacyclic group is the extension of a cyclic group by a cyclic group) transitive permutation group of degree $n$? It can be easily proved that ...
1
vote
0answers
104 views
simply primitive permutation groups of degree $2p^2$
I know that the only simply primitive permutation groups of degree $2p$, where $p$ is an odd prime, are $A_5$ and $S_5$. I want to know that: Is there a complete list of simply primitive permutation ...
16
votes
1answer
847 views
Order of products of elements in symmetric groups
Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying
$1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$
whose product has order ...
3
votes
1answer
193 views
automorphism group of orbital graphs
Let $G$ be a transitive group on $\Omega$. Every orbits of $G$ on its natural action on $\Omega\times\Omega$ is called an orbital of $G$ on $\Omega$. For each orbital $\Delta$ of $G$ on $\Omega$, the ...
5
votes
2answers
378 views
Permuting Racked Pool Balls with a Single Break
Given reasonable physical assumptions (on friction, collisions, etc.), would it be possible to "break" in a pool game such that when all the balls come to rest, the only difference is that the racked ...
0
votes
0answers
109 views
simple graphs of degree 16 with a semiregular normal subgroup isomorphic to the quaternion group $Q_8$
Is there any simple graph $\Gamma$ with 16 vertices with full automorphism group $G$ such that $H\cong Q_8$ be a semiregular normal subgroup of $G$?
0
votes
1answer
204 views
vertex-transitive graphs of order 10 with full automorphism group $A_5$ or $S_5$.
By a well-known result we know that a simply primitive permutation group of degree $2p$ where $p$ is a prime is $A_5$ or $S_5$ acting on 2-subsets of $\{1,\ldots,5\}$. The group has rank 3 and the ...
0
votes
1answer
61 views
imprimitive 2-blocks in connected Cayley (di)graphs of order twice a prime
Let $\Gamma=Cay(G,S)$ be a connected Cayley (di)graph over a group of order twice a prime and $\Sigma$ be a complete system of 2-blocks for $Aut(\Gamma)$. Let $K$ be the kernel of the action of ...
4
votes
0answers
138 views
A conjecture on Zassenhaus groups
In a 1995 paper by Ali Nesin, "Permutation groups of finite Morley rank", the following conjecture is mentioned:
Conjecture 5 An infinite Zassenhaus group $G$ of finite Morley rank is isomorphic ...
9
votes
0answers
449 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 ...
2
votes
0answers
202 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
172 views
Outer automorphisms 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 ...
0
votes
2answers
191 views
Semiregular subgroups of automorphism group of cayley graphs
Let $\Gamma$ be a Cayley graph over group $K$ and $H$ be a semiregular subgroup of $Aut(\Gamma)$ with two orbits. Then $|K|=2|H|$. Is there any other relation between $H$ and $K$ in general? What ...
4
votes
1answer
232 views
polycirculant conjecture
By the polycirculant conjecture, every vertex-transitive graph is a polycirculant graph (D. Marusic 1981 and D. Jordan 1988).
There are two papers that claim to prove this conjecture:
1. A. Golubchik, ...
8
votes
3answers
486 views
partly obscured Rubik's cube
I just came back from a beach which features a large Rubik's cube (2m high). The base of the cube is not visible and the top is not coloured. The four vertical sides are each divided $3\times 3$ into ...
7
votes
1answer
277 views
Maximal subgroups of a certain finite 2-group
The following came up in a problem on reconstruction of digraphs. I determined enough about the answer to satisfy the application completely, but still I am curious to know what the complete solution ...
1
vote
2answers
199 views
Identifying Subgroups of the Modular Group via Permutation Representations on Cosets
Suppose we have a known group G and an unknown subgroup H. The permutation representation of G on the cosets of H gives a permutation group C, which is known. Is it possible to identify the generators ...
2
votes
0answers
232 views
Does probability of a derangement go up under passing to subgroups? [closed]
This is prompted by my attempts to work on this question. Let $H \subset G \subseteq S_d$ be transitive permutation groups. Recall that an element of $S_d$ is called a derangement if it has no fixed ...
2
votes
2answers
193 views
Mclaughlin Graph
how can i construct a strongly regular graph with parameter $(275,112,30,56)$(Mclaughlin Graph), (105,32,4,12)?
I need adjacency matrix of them?
I know they are unique.
4
votes
1answer
232 views
Cardinals of transitive permutation groups acting on $\{1,\dots,n\}$
Is there a nice description of all divisors of $n!$ which can be realized as cardinals of permutation groups acting transitively on $\{1,\dots,n\}$?
A necessary condition is of course that such a ...
6
votes
2answers
488 views
Two groups acting on a set.
Suppose we are given a set S of points on which two different groups G and G' (given by sets of generating permutations) act. Is there an efficient algorithm for finding generators the largest pair ...
11
votes
2answers
370 views
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 ...
3
votes
1answer
413 views
Alternative Definition of the Quantum Determinant?
Let $M_q(n)$ be the standard quantum matrices (over the complex numbers) with generators $u^i_j$ for $i,j = 1, \ldots ,N$, and reations
$$
u^i_ju^k_j = qu^k_ju^i_j, \text{ for } i < k, ~~~~~~~ ...
4
votes
1answer
369 views
Symplectic groups Sp_{2m}(2) as 2-transitive permutation (i.e. Galois) groups
Hello,
I am looking for information about the symplectic groups $Sp_{2m}(2)$ as permutation group acting on quadratic forms.
Consider the block matrices
...
5
votes
1answer
291 views
Counting the (additive) decompositions of a quadratic, symmetric, empty-diagonal and constant-line matrix into permutation matrices
Dear community,
I have the following combinatorial question which I will explain in short first and then with some more detail. At the end you will find a very simple example.
Short version
Le $A ...
3
votes
1answer
632 views
The number of orbits of a permutation action
Let $G$ be a finite group acting on a finite set $\Omega$. A general question is to determine the sequence $o_k(\Omega)$, where $o_k(\Omega)$ is the number of orbits on $G$ for the natural action of ...
2
votes
1answer
148 views
Name for a certain kind of 'weakly primitive' permutation group
I have found it necessary to define the following property:
Consider a finite set $X$ and a group $H$ of permutations of $X$. Suppose for every normal subgroup $K$ of $H$ that $K$ acts faithfully on ...
