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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If $G = R \rtimes G_{\alpha}$ and $R$ is non-abelian of order $27$ and exponent $3$, then $G_{\alpha} \ncong D_{12}$

Suppose that $G = R \rtimes G_{\alpha}$ is a finite permutation group on $\Omega$ where $R$ is the non-abelian subgroup of order $27$ and exponent $3$ (see here). Suppose that $G_{\alpha} \cong D_n$, ...
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An example of lifting a group action to the universal cover.

Through a previous question, I understood how we can lift the action of a group $G$ on a topological space $X$ to an action of a covering group $G'$ of $G$ on the universal cover $\tilde{X}$ in such a ...
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Finding a map from $X = (0,\infty) \times (0,\infty)$ to a cone

Determine the quotientspace $X / \Gamma$, where $\Gamma = <\phi>$, $\phi(x,y) = (x/2,2y)$ and $X = (0,\infty) \times (0,\infty)$. I think the quotient space has to be a cone, but I can't figure ...
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Properly discontinuous action of a group

Let $\Gamma=\{\varphi^n\mid n\in\mathbb{N}\}$ where $\varphi(x,y)=(\frac{x}{2^n},2^ny)$. I am trying to decide if $\Gamma$ defines a properly discontinuous action on $X=(0,\infty)\times(0,\infty)$. I ...
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Is the axiom $g1 = g$ essential for a group action

A group $G$ acts on a set $\Omega$ if (1) $\omega\cdot 1_G = \omega$ (2) $(\omega \cdot g)\cdot h = \omega \cdot (gh)$ for all $\omega \in \Omega$ and $g,h \in G$. But is (1) really essential, ...