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0
votes
0answers
97 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 ...
6
votes
1answer
205 views

Primitive, non-2-transitive groups with very large orbitals?

Let $G$ be a transitive permutation group on a set $X$ with $n$ elements. Assume that $G$ is primitive, i.e., $G$ preserves no non-trivial partition of $X$. Assume as well that $G$ is not ...
3
votes
0answers
52 views

Double coset relation for unique intermediate subgroup (with homogeneity)

Let $G$ be a group and $H$ a subgroup such that there is a unique (non-trivial) intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$) and $(H \subset K) \sim (K \subset G)$ (homogeneity) ...
2
votes
0answers
91 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
158 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
119 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
275 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
163 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
166 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
155 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
245 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
173 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
433 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
52 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
189 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
187 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
124 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
337 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
101 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
350 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
188 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
155 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
150 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
908 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
201 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
388 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
110 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
214 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
62 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
139 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 ...
10
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0answers
462 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
203 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
173 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
195 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
235 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
495 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
282 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
202 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
234 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
194 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
233 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
490 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
371 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
427 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
390 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
293 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
657 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 ...