# Questions tagged [group-actions]

Use with the (group-theory) tag. Groups describe the symmetries of an object through their actions on the object. For example, dihedral groups of order $2n$ act on regular $n$-gons, $S_n$ acts on the numbers $\{1, 2, \ldots, n\}$ and the Rubik's cube group acts on Rubik's cube.

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### $3$-length of $(C_3\times C_3):GL(2,3)$ in GAP

I try to find $3$-length of the semidirect products group $(C_3\times C_3):GL(2,3)$. $p$-length means the number of factors in the shortest subnormal series which factors are $p$-groups or $p'$-groups....
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### How to Find Orbits and Stabilizers

The question is: Let $S$ denote the set of possible black-or-white colorings of the edges of an equilateral triangle. The triangle's symmetry group $D_6$ acts naturally on $S$. (a) How many orbits are ...
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### Prove new group action is properly defined. [closed]

In continuation with part (b) of the problem here. Show that there is injective map from the sets of actions on set $X$ to set of new-action on set $Y.$ How to proceed?
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### Show that a Modified-group action, is also a group action.

Let $G$ be a group and $X$ a set. Define a modified-action as for which identity may not hold. Have a subset $Y$ of the set $X$ where holds:$\,e\star x\mid x.$ Can it be assumed that $e\star y$ has ...
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### Classification actions of finite groups

This is a very elementary question, but I'm having trouble finding anything explicit about it: Consider a finite group $G$ with an action on a finite set $X$. Is there a classification of such actions ...
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### Find sizes of disjoint subsets to make transitive action by icosahedron on set $X$.

Could only prepare ground for this in my last post here, with a reduced group and set size. Kindly give hints, as sizes of subsets that can think of are too small: $X_3.$ Though read of tetrahedron ...
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### Difference between group action and transitive group action. [closed]

These are my notes, request vetting and help in doubts. Made to understand problem stated at: Find sizes of disjoint subsets to make transitive action by icosahedron on set $X$. Let us start with an ...
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### How close is $GL_n (2)$ to being $3$ transitive?

The group $GL_n(2)$ acts transitively on the $2^n-1$ nonzero vectors of $\mathbb{F}_2^n$. This action is in fact 2-transitive since any pair of distinct nonzero vectors in $\mathbb{F}_2^n$ is ...
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### Specific questions regarding the group action $G \times \mathcal{S}(G) \to \mathcal{S}(G)$

Problem statement: Let $G$ be any group and recall $\mathcal{S}(G)$ denotes the set of subgroups of $G$. Let $G \times \mathcal{S}(G) \to \mathcal{S}(G)$ be given by $$g \cdot H = gHg^{-1}$$ Prove ...
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### Compute the number of distinct actions of cyclic group $C_n$ on a set $X,$ s.t $|X|= n+1.$

There are $n+1C_n = n+1$ combinations possible, with $n!/n= (n-1)!$ orderings possible in each; leading to a total of $(n+1).(n-1)!$. Let $n= 6.$
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### Number of different ways to get permutations of disjoint cycles of given length.

This question is derived (as want to derive the formula for below problem, and also general approach) from : Compute the number of distinct actions of $C_m$ on set $X,$ s.t. $|X|= n= 2m+1.$ Let ...
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### When is multiplication by an element from a transformation group bijective?

I am reading Allan Clark's Elements of Abstract Algebra and he states the following about transformation groups. We note that each $g \in G$ determines a one-to-one correspondence $g: X \to X$ given ...
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### Actions on Bernoulli space have almost the same orbits

Let $(\mathbb{B}, \nu)$ be the binary space with probability measure $\nu(0) = \nu(1) = \frac{1}{2}$. The map $T : \mathbb{B}^{\mathbb{N}} \to \mathbb{B}^{\mathbb{N}}$ is defined as left addition with ...
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### Finding stabilizer and orbit of power set of a group

Let $H$ be a subgroup of a finite group $G$. Let $A$, $B$ $\in \mathcal{P}(G)$. Define $A$ to be conjugate to $B$ with respect to $H$ if $B=hAh^{-1}$ for some $h\in H$. Then find orbit and stabilizer ...
I know that a group action is transitive when there is one orbit. Say that $G$ is a group acting on the set $A$. The identity element of $G$ will clearly create $|A|$-many orbits. But the other ...