# 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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### Every group of order $60$ , having a normal subgroup of order $2$ , has a normal subgroup of order $6$ (without Sylow )?

How to prove , without using Sylow's theorems , that every group of order $60$ , having a normal subgroup of order $2$ , contains a normal subgroup of order $6$ ? Please help . Thanks in advance
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### Let $G$ be a permutation group and $R \unlhd G$. If $G$ acts double-transitive on the orbits of $R$, then $G / R \cong A_5$ and we have $5$ $R$-orbits

Let $G$ be a transitive permutation group such that every nontrivial element fixing some point fixes exactly three points. Also suppose that $G_{\alpha} \cap G_{\beta}\cap G_{\gamma} < G_{\alpha}$ ...
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### More question on the proof of orbit-stabilizer theorem from Gowers's weblog

Still I'm reading Gowers's weblog about orbit-stabilizer theorem, I must admit that my understanding of this materiel improved, but still I have some question. Let $G$ be a finite group, and $X$ be ...
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### Show that $|C_G(u)| = 12$ by “counting involutions”

Let $G$ be a transitive permutation group acting on $\Omega$ such that every non-trivial element fixing some point has exactly three fixed points. Suppose $G_{\alpha} \cong A_5$ for some point ...
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### If $G = VN$, $V$ a four group and $N$ regular normal, then there exists some Sylow subgroup left invariant by $V$

Let $G$ be a permutation group on $\Omega$ with $G = VN$, where $V \cong C_2 \times C_2$ (the four-group) and $N$ has odd order with some prime divisor $>3$. Suppose $N$ is a regular normal ...
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### Groups of prime power and the fixed point set

Suppose that $X$ is a finite $G$-set. A group $G$ is of prime power if $|G|=p^n$ for $p$ prime. The fixed point set $X_G=\{x\in X : gx=x$ $\forall g\in G\}$. I'm asked to prove that $|X|=|X_G|$ (mod ...
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### A subtle error in a “change of variable” in a sum

Let $G$ be a finite group of order $n$, and let $E$ be a finite set. Let $\star$ be an action of $G$ on $E$. Suppose that $G \star x_1,..., G \star x_m$ are the distinct orbits of elements in $E$. ...
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### If $G_{\alpha} \cong S_4$ and $|\mbox{fix}(g)| \in \{0,3\}$ for $g \ne 1$. Then $G$ has transitive normal subgroup of index $2$.

Let $G$ be a transitive permutation group such that $|\mbox{fix}(g)| \in \{0,3\}$ for every nontrivial $g \in G$. Also suppose $|N_G(G_{\alpha}) : G_{\alpha}| = 1$, i.e. $G_{\alpha}$ is the only fixed ...
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### Given $\pi:X\rightarrow Y$ how to show $X$ is irreducible (resp. normal) $\Rightarrow$ $Y$ is irreducible(resp. normal)?

Let $G$ act on the affine variety $X=\operatorname{Spec}(R)$ such that $R^G$ is a finitely generated $\mathbb C$ - algebrs and let $\pi:X\rightarrow Y=\operatorname{Spec}(R^G)$ be the morphism of ...
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### Transitive action on a finite set and group

If $G$ is a finite group and acting transitively on a set $X$ with $|X|>1$. then I have two question :- There is some element of $G$ in which fixes no element of $X$. Give a counter-example to ...
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### If $G_{\alpha} \cong A_4$ and $|\mbox{fix}(g)| \in \{0,3\}$ for $g \ne 1$ and $V \le G_{\alpha}$ is the four-group in $A_4$, then $C_G(V) = V$

Let $G$ be a transitive permutation group such that $|\mbox{fix}(g)| \in \{0,3\}$ for every nontrivial $g \in G$. Also suppose $|N_G(G_{\alpha}) : G_{\alpha}| = 1$, i.e. $G_{\alpha}$ is the only fixed ...
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### Why is this a group action - what is the significance of $g^{-1}$?

Let $G$ be a group acting on a variety $X$ such that every $g\in G$ defines a morphism $\phi_g:X\rightarrow X$ given by $\phi_g(x)=g\cdot x$. If $X=\operatorname{Spec}(R)$ is affine then $\phi_g$ ...
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### If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $N \unlhd G$. Then $G_{\alpha}N$ is normal if we have $p$ orbits of $N$.

Let $G$ be a transitive permutation group acting on $\Omega$ such that each nontrivial element either has no fixed point or exactly $p$ fixed points. Suppose that for $g \notin N_G(G_{\alpha})$ we ...
### If $G$ is solvable and acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $M$ is maximal normal. Then $|G/M| = p$.
Let $G$ be a transitive permutation group on $\Omega$ which fulfills the following property (P) (P) each nontrivial element fixes no point or exactly $p$ points. for some odd prime $p$. Further ...