# Tagged Questions

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### Prove this wreath product is a group [Homework]

I'm not usually one to post unworked problems here... I usually try to at least have an attempt, but unfortunately in this case I'm unable to even get an intuitive sense of what's going on here - and ...
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### Any ring is integral over the subring of invariants under a finite group action

I need to prove that if $G$ is a finite group that acts on ring $A$, and $A^G$ is the subring consisting of elements of $A$ which are invariant under all $g\in G$, then $A$ is integral over $A^G$. ...
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### Can we prove $H \cong xHx^{-1}$ given $H \le G, x \in G$ using group action?

The exercise is as follows: $G$ is a group, $H \le G$. For any $x \in G$, to prove that $H \cong xHx^{-1}$. I am able to prove this isomorphism by defining a bijection $f : h \mapsto xhx^{-1}$ ...
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### Find the orbit space $T^2 / \mathbb Z_2$

Let $T^2$ be the unit torus $$T^2 = \left\{ (\lambda, \lambda') \in \mathbb C^2 \mid |\lambda| = |\lambda'| = 1 \right\}.$$ Then the group $\mathbb Z_2$ is acting on $T^2$ by the rule ...
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### Classification of transitive G-sets for a given group of small order

Given a group of small order (<30), how does one go about systematically finding all the transitive G-sets up to isomorphism? By X and Y being isomorphic we mean there are maps $f:X \rightarrow Y$ ...
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### Stabilizer map on transitive G-set defines a morphism with G acting on subgroups by conjugation

This is part of a homework problem for a graduate course on abstract algebra. Given a transitive G-set $X$, show that the map that assigns to $x \in X$ its stabilizer defines a morphism of G-sets ...
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### Group acting by isometries on a length space

I am reading the book A course in metric geometry by Burago, Burago and Ivanov. I have some difficulties with an exercise 3.4.6 on page 78. The exercise is the following: Let a group $G$ act by ...
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### Proving those elliptic matrices in $\operatorname{SL}_2(ℤ)$ are not conjugate

Set $\mathbf{\Gamma} = \operatorname{SL}_2(ℤ)$, let $\mathbf{H}$ denote the upper half plane. and let \Gamma_0 (N) = \left\{ \left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right] ...
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### Group acting on a set

Let $A$ be a set, and let $G$ be any subgroup of $S_A$. $G$ is a group of permutations of $A$; we say it is a group acting on the set $A$. Assume here that $G$ is a finite group. If $u \in A$, the ...
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### Studying the action of $GL(V)$ on the vector space $V$

The statement I am trying to prove is the following. Let $k$ a field and $V$ a $k$-vector space of finite dimension. Let $\mathscr{B}$ be the set of ordered $k$-bases of $V$. The natural ...
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