1
vote
2answers
39 views

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 ...
1
vote
1answer
55 views

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 $A^G$. ...
0
votes
0answers
214 views

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 $f : h \mapsto xhx^{-1}$ ...
0
votes
0answers
33 views

Action of a Lie group, a map of constant rank

Consider some Lie group $G$, smooth manifold $X$ and some action of $G$, i.e. a group homomorphism $\mathcal{A}: G\longrightarrow \mathrm{Diffeo}(X)$ such that the map $(g,x)\mapsto ...
0
votes
2answers
37 views

Embedding monomorphism between Symmetric Groups

Suppose that $m$ and $n$ are positive integers, and $m<n$. Define $I:S_m \rightarrow S_n$ as follows: Given $\alpha \in S_m$, we let $\hspace{150pt}I(\alpha)(k)=\alpha(k) ...
1
vote
0answers
53 views

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 ...
2
votes
1answer
137 views

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 $f:X \rightarrow Y$ ...
2
votes
0answers
92 views

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 ...
2
votes
1answer
66 views

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 ...
2
votes
2answers
99 views

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] ...
-1
votes
1answer
70 views

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 ...
1
vote
1answer
52 views

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 ...
3
votes
1answer
108 views

Show that the orbits of $S_n$ under the conjugation action of $S_n$ on itself correspond 1-1 with the cycle types.

Show that the orbits of $S_n$ under the conjugation action of $S_n$ on itself correspond 1-1 with the cycle types. So, the orbit of $\sigma \in S_n$ is the set $S_n \sigma = \{ \tau .\sigma : ...
0
votes
2answers
502 views

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 ...
3
votes
1answer
159 views

On Group action and blocks of subgroups of the symmetric Group

this exercise is from Dummit and foote , page 117 , # 7.d prove : a transitive group $G$ is primitive on $A$ iff for each $a \in A$ , the only subgroups of $G$ containing $G_a$ are $G_a$ and $G$ ...
3
votes
3answers
321 views

Exercises about group actions

I am having trouble with exercises from Chapter IV of Aluffi's Algebra: Chapter 0. After spending many hours on the following two, I guess I need some help: Let $G$ be a finite group, and suppose ...
1
vote
2answers
116 views

Composite group homomorphism between alternating groups

Let $N$ a non-trivial normal subgroup of $A_n$ and $H = N \cap A_{n-1}$. I would like to show that $A_{n-1} \hookrightarrow A_n \to A_n/N$ is surjective, where $A_n \to A_n/N$ is the canonical ...
3
votes
2answers
265 views

Transitive action of normal subgroup of the alternating group

everyone! Would anyone be willing to give me any sort of help with the following question? Let $n\ge 4$ and $A_n$ the alternating group. Let $N$ a non-trivial normal subgroup of $A_n$. Prove that the ...
4
votes
0answers
239 views

Semi-direct product isomorphic to direct product

I would like some help on the following problem from anyone who would like to help. Let $f: H \to G$ be a group homomorphism. For $h \in H$, define $\rho(h) = \phi_{f(h)} \in Aut(G)$. The situation ...
0
votes
3answers
94 views

A step in the proof of Cauchy's theorem for groups

Let $G$ a group, $p$ a prime and $X=G^p$. Let $\sigma\in S_X$ act as follows: $\sigma(x_1,...,x_p) = (x_2,...,x_p,x_1)$. Let $Y$ the subset of elements in $X$ such that $x_1x_2...x_p=1$ and let ...
0
votes
1answer
121 views

Injective Homomorphism on $S_n$

I have the following question on a homomorphism between symmetric groups and it has been really a pain. I know I am supposed to use induction but, I seem to miss something essential so if anybody ...
1
vote
1answer
63 views

Showing equality of two groups.

Consider $S_n$, the permutation group. Let $a\in S_n$. I want to show that if $a$ is an $n$ or an $n-1$ cycle, then $\langle a\rangle = \{c \in S_n : ca=ac\}$. Any help will be veru much appreciated.
0
votes
1answer
68 views

A question on groups actions of permutation groups

This is the final step into completing a problem and I am a bit stuck. I need to show that: Consider the action of $S_{n-1}$ on $S_n/S_{n-1}$ by left multiplication. Does this action have exactly two ...
1
vote
1answer
354 views

How to find isomorphism classes of transitive actions

I have not been able to find a proper definition of what an isomorphism class is (in the context of group theory). If one could define it properly for me and give me some help with the following two ...
0
votes
0answers
109 views

Questions (doubts) on: Group Action on Manifolds

There are 2 questions that are bugging me in differential topology and I'd be glad if the same could be cleared up: Let $X = x\frac{\partial}{\partial y}$ be a vector field on $M = R^2$, where $R$ ...
3
votes
2answers
552 views

Algebra - Infinite Dihedral Group

Let $G$ be the set of bijections $\mathbb{R} \to \mathbb{R}$ which preserve the distance between pairs of points, and send integers to integers. Then $G$ is a group under composition of functions. The ...
0
votes
0answers
64 views

G-H biset and left & right G-set

1) Prove that each left $G$-set $X$ can be turned into a right $G$-set by defining $x\sigma = \sigma^{-1}x$, and that every right $G$-set arises in this way. I have the left action $X \times G \to X$ ...
3
votes
1answer
211 views

Extending a group action to a quotient group

If I have a cyclic group $G = (a)$ acting on an abelian group $A$, I need to define a natural action of $G$ on the quotient space $A/B$, where $B$ is a normal subgroup of $A$ with the property that ...