The permutation-groups tag has no wiki summary.

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### Number of Different Combination Possible? [migrated]

You are given number of places as m and number of digits as n.You have to fill those m places in such a way that each digit appears atleast one time.
For Example
Given m as 4 and n as 3
so you have ...

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### Confusion about the projected component in an irreducible space in the tensor product decomposition using Littlewood-Richardson?

The regular representation of the symmetric group can be formulated in terms of an abstract tensor, where the action of the symmetric group elements is by means of permuting the indices. Given an ...

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### Classification of indecomposable inclusions $(H \subset G)$ with $G$ decomposable

Definition: A group $G$ is indecomposable if: $G = G_1 \times G_2 \Rightarrow \exists i \ G_i = 1$.
We can generalize the notion of indecomposable from groups to inclusion of groups as ...

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### 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: ...

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### What's the ratio of inclusions of finite groups with a distributive lattice?

Definition: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the normal core of $A$ in $B$.
...

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### Generalization of Frobenius groups

Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point.
In other words, if in a ...

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### When is the product of two permutations cyclic?

Call a subset $H \subset S_n$ "simple" if for every $q \in S_n$, there is some $p \in H$ such that $pq$ is cyclic (i.e. consists of a single cycle of length $n$).
Is there some characterization known ...

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### 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 ...

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### When is $S_n \times S_m$ a subgroup of $S_p$?

I asked the following question on math.stackexchange several months ago:
Let $n,m,p>1$ be such that $S_n \times S_m \hookrightarrow S_p$. Does it imply that $p \geq n+m$?
Derek Holt gave a ...

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### Question about doubly transitive groups with an n-cycle

Let $G$ be a doubly transitive subgroup of $S_n$ which contains an $n$-cycle, and let $G_{12}$ be the subgroup of $G$ consisting of all elements $g\in G$ for which $g(1)=1$ and $g(2)=2$. Then ...

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### 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 ...

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### 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 ...

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### 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) ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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: PÃ¡lfy, P. P.
A polynomial bound for the orders of primitive solvable groups.
J. ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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
...

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### 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;
...

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### the symmetric group $S_{2^{râˆ’1}}$

Is there any routine technique to find a set of permutations which generate a Sylow 2-subgroup of the symmetric group $S_{2^{râˆ’1}}$?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $G$?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### Outer automorphisms of an infinite simple group

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...