The permutation-groups tag has no usage guidance.

**2**

votes

**5**answers

262 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

**2**answers

241 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

**0**answers

191 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

**1**answer

196 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

**2**answers

791 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 ...

**1**

vote

**1**answer

61 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 ...

**12**

votes

**0**answers

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
...

**8**

votes

**1**answer

344 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

**2**answers

141 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 $S_{2^{r−1}}$?

**5**

votes

**3**answers

478 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

**0**answers

118 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

**1**answer

658 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 ...

**4**

votes

**2**answers

279 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?

**2**

votes

**2**answers

336 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

**2**answers

168 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

**0**answers

112 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 ...

**23**

votes

**2**answers

2k 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 ...

**4**

votes

**1**answer

290 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

**2**answers

540 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 ...

**2**

votes

**1**answer

159 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

**1**answer

291 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

**1**answer

73 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

**0**answers

177 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 ...

**4**

votes

**0**answers

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 ...

**15**

votes

**0**answers

642 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

**0**answers

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

**0**answers

190 views

### Outer automorphisms of an infinite simple group

**0**

votes

**2**answers

229 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

**1**answer

297 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

**3**answers

554 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

**1**answer

410 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

**2**answers

260 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

**0**answers

260 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 ...

**3**

votes

**3**answers

273 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

**1**answer

246 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

**2**answers

523 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 ...

**20**

votes

**3**answers

669 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

**1**answer

596 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, ~~~~~~~ ...

**5**

votes

**1**answer

770 views

### Symplectic groups $Sp_{2m}(2)$ as $2$-transitive permutation (i.e. Galois) groups

I am looking for information about the symplectic groups $Sp_{2m}(2)$ as permutation group acting on quadratic forms.
Consider the block matrices
$$e=\begin{pmatrix}0&1\\0&0\end{pmatrix}, ...

**6**

votes

**1**answer

328 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

**1**answer

1k 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

**1**answer

166 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 ...

**10**

votes

**4**answers

2k views

### An easy proof that S(n) does not embed into A(n+1)?

Rotman's book An Introduction to the Theory of Groups (Fourth Edition) asks, on page 22, Exercise 2.8, to show that S(n) cannot be embedded in A(n+1), where S(n) = the symmetric group on n elements, ...