Tagged Questions

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 $2n$ acts 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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How to define own group action in GAP?

I am beginner in GAP. I have a group and a set. I wish to define an action the group on the set in my own way and wish to calculate its orbits and stabilizers. Is it possible? What is process?
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computing the orbits for a group action

Let $G$ be the Galois group of a field with nine elements over its subfield with three elements. Then the number of orbits for the action of $G$ on the fields with nine elements is 3 5 6 9 I have ...
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On Conjugacy Classes of Alternating Group $A_n$

In Dummit & Foote, page 131 Let $K$ be a conjugacy class and suppose that $K$ is subset of $A_n$ . Show that if $\sigma$ belongs to $S_n$ then , $\sigma$ does not commute with any ...
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Motivation for the term “transitive” group action

I have two questions: In a text, I read that a group permutes pairs of faces of a solid transitively. Geometrically, what are they referring to, and what is an example of when a group may not ...
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Properties of set $\mathrm {orb} (x)$

Properties of set $\mathrm {orb} (x)$: ${\displaystyle \bigcup_{x\in X}\mathrm{orb}(x)=X}$; $\mathrm{orb}(x)\cap\mathrm{orb}(y)=\emptyset$ for all $x,y\in X, x\neq y$ How to prove it? Please help....
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If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$, $M$ maximal with $|G : M| = p$, then $|M / L| = p$ for semiregular $L \unlhd G$.

Let $G$ be a solvable, nonregular and transitive permutation group acting on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points for some prime $p$. And suppose that ...
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Let $G$ act such that $|\mbox{fix}(g)| \le 2$ for $g \ne 1$. Then the Sylow $2$-subgroups acts regular on certain orbits
Let $G$ be a finite permutation group on $\Omega$ acting nonregular and transitive such that each nontrivial element fixes at most two points of $\Omega$. Suppose that for $\alpha \in \Omega$ the ...
Let $p$ be a rational prime and $H$ be a finite cyclic group of prime order $l$ prime to $p$, i.e. $(l,p) = 1$. Let $G$ be a finite abelian group of $p$-power order. If I can write an (abelian) group ...