3
votes
4answers
43 views

Prove a property about the centralisator

Let G be a group and $U \subseteq G$ a subgroup. Let $x \in G$ be arbitrary. How to show that $C_G(xUx^{-1})=xC_G(U)x^{-1}$ where $C_G(U):=\{g\in G : gu=ug$ $\forall u\in U\}$ For the first ...
0
votes
0answers
32 views

Must the index $k=|G:HC_G(x)|$ be finite?

I want to solve the following Exercise from Dummit & Foote's Abstract Algebra text: Assume $H$ is a normal subgroup of $G$, $\mathcal{K}$ is a conjugacy class of $G$ contained in $H$ and $x ...
1
vote
1answer
41 views

Double cosets and conjugation

Let G be a group and $h,g \in G$ with $SgT=ShT$ Show that the subgroups $gTg^{-1}\cap S$ and $hTh^{-1}\cap S$ are conjugated in S. Two subgroups $U,V$ are conjugated if $\phi(U)=gUg^{-1}=V$, right? ...
1
vote
1answer
29 views

For $f \in \mathbb{R}[T_1,\dots,T_n]$ and $\sigma \in S_n$, the number of different polynomials of the form $\sigma(f)$ divides $n!$

I have the following problem that I believed I solved but am trying to understand better. Let $n\geq 2$. Consider $\sigma \in S_{n}$ and $f \in \mathbb{R}[T_1,\dots , T_n]$, let ...
0
votes
1answer
47 views

The action of free group on line

Let $G$ be a free group, if the action of $G$ on $\mathbb{R}$ is free, does it imply that $G$ is abelian?
1
vote
1answer
32 views

The function $A:\mathbb{Z}\times\mathbb{R}\to\mathbb{R}$ given by $(n,x)\to nx$ is a group action on $\mathbb{R}$.

I somehow came through this True or False question, but before answering it, I just wanted to ensure some things. I understand that the axioms for $group-action$ must be met. Basically: ...
2
votes
1answer
33 views

Understanding the structure of a module over a group algebra

Suppose one has a permutation group $G$ acting on the set $[n] = \{1, 2, \ldots, n\}$, which extends naturally for any field $F$ to a $FG$-module structure on the set $F[n]^k$ of formal $F$-linear ...
1
vote
1answer
24 views

Intersection of stabilisers

I have a short question: if $G$ acts on $X$, is the intersection of all the stabilisers the same as the conjugates of the stabilisers? In other words does, for any $z\in X:$ $$\bigcap_{g\in G} ...
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 ...
0
votes
0answers
39 views

Representation theory& module

$V$ is a left $R$ module, how do you understand the ring homomorphism $$\rho_{V}:R \to End_Z(V)$$ I know that it is like a group acting on sets, but it is very easy to understand like a group $S_n$ ...
3
votes
1answer
84 views

Why it is a group action?

Let a group $G$ acts on a vector space $V$ and let $f$ be a function on $V$. The action of an element $g \in G$ defined by the rule $g f(x)=f(g^{-1} x), \forall x \in V.$ A typical proof from a ...
2
votes
1answer
36 views

How many orbits are there in the group action $A\colon 3\mathbb{Z} \to\mathbb{Z}_{6}$ with action given by $(3n,m)=(3n+m)\bmod6$.

I am having difficulty trying to understand this question. All I know is that $3\mathbb{Z}$ and $\mathbb{Z}_6$ are both groups, that is: $$3\mathbb{Z}=\{\dotsc, -6, -3, 0, 3, 6 \dotsc\}$$ ...
0
votes
0answers
50 views

Group action problem.

I've formed up a group $G=\{(1),(34),(12)(34),(124)\}$ acting on a set $X= \{1,2,3,4\}$. Knowing the axioms for a group action, that is: (Compatibility with identity): $e*x=x$ for all $x\in X$ ...
0
votes
0answers
42 views

How many orbits are there in the group action of $S_4$ on $X={1,2,3,4}$?

So far I know that the order of the group is 4! which is 24 elements. I know that X the set has 4 elements. Finally, I understand that the Orbit-Stabilizer theorem states that: ...
2
votes
1answer
80 views

Representation of $GL_2$ on $K^2$

In one of my problems it says the following: Let $K$ be an infinite field. Consider the linear action of $GL_2$ on $K[x,y]$ induced by the natural representation of $GL_2$ on $K^2$. I don't know what ...
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
221 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
1answer
23 views

Compute $S_3$ acting by conjugation on the set $X$ of $6$ subgroups of $S_3$

I know that the subgroups of $S_3$ are $\{e\}$, $\langle(12)\rangle$, $\langle(13)\rangle$, $\langle(23)\rangle$, $A_3$, and $S_3$. What I also know is that conjugation is $C_g(H) = gHg^{-1}$. Thus in ...
1
vote
0answers
45 views

Notation for pointwise versus “setwise” stabilizers

Suppose one is working with both pointwise and setwise stabilizers of sets under a group action. Are there common conventions for notationally distinguishing these two notions? How common are they? ...
2
votes
1answer
90 views

Does $GL(n,K)$ act transitively on $1$-dim subspaces of $K$

If we let $K$ be a field and $GL(n,K)$ act by right multiplication on the $1$-dim subspaces of $K^n$. Then if we take $\langle v_1 \rangle, \ldots \langle v_n \rangle \in K^n$ distinct and $\langle ...
1
vote
1answer
24 views

a question concerning subgroup of symmetric group

Suppose $H$ is a transitive subgroup of the symmetric group of $n$ symbols. Show that $n$ divides the order of $H$. I tried to show that some $n$-cycle is in $H$ but this idea did not work.
0
votes
2answers
73 views

Show that group action is homomorphism to Symmetric group

I'm just barely getting my feet wet with abstract algebra, currently working on understanding group action. According to the wikipedia article, a group action $A$ of group $G$ on set $X$ is a group ...
5
votes
5answers
307 views

Poincaré's theorem about groups

Let $G$ be a group and $H<G$ such that $[G:H]<\infty$. There exists a subgroup $N\triangleleft G$ such that $[G:N]<\infty$. I have to show this fact (that according to my book is due to ...
2
votes
1answer
55 views

Orbits that 'coalesce'

Let $R$ be a commutative ring, $G$ a group scheme over $\mathrm{Spec}\;R$, and $X$ a scheme over $\mathrm{Spec}\;R$ on which $G$ acts $R$-morphically via $G\times X\to X$. Suppose $S$ is another ...
1
vote
1answer
85 views

What are the conjugacy classes in $\mathrm{Aut}(G)$?

Let $G$ be an arbitrary group, and let $\mathrm{Aut}(G)$ be the group of automorphisms of $G$ (with composition of morphisms as multiplication). I'd like to learn more about the problem of ...
1
vote
1answer
44 views

What motivates the definition of “Periodic” group action

Consider a group $G$ acting on a set $\Omega$. For example, let $G=\{g\in A(\mathbb R):(\alpha +1)g=\alpha g+1\}$ for all $\alpha\in\mathbb R$, where $A(\mathbb R)$ are the order-preserving ...
1
vote
1answer
79 views

Group action on set of maps - formula

It is given that $G:X$ and $G:Y$. Does this $[g\bullet f](x) := g\bullet f(g\bullet x)$ formula define group action $G:(Y^{X})$ I guess it doesn't, but I can't prove it as for now. And there must be ...
0
votes
0answers
30 views

Finding conjugacy classes of $D_{10}$

Looking at the group $D_{10}$, I have found that for some (non-identity) rotation $\rho$ its centraliser has order 5, and for some reflection $\tau$ its centraliser has order 2. By the ...
3
votes
0answers
45 views

Partial order on the orbits of the variety of commuting nilpotent matrices

The variety of nilpotent $n\times n$ matrices $\mathcal{N}_n$ over an algebraically closed field $k$ is the disjoint union of orbits under the action of conjugation by $GL_n(k)$. These orbits are ...
0
votes
4answers
109 views

Prove that the number of elements of every conjugacy class of a finite group G divides the order of G.

Prove that the number of elements of every conjugacy class of a finite group $G$ divides the order of $G$. I'm studying for my Group Theory exam and this was a question on a previous exam. I ...
1
vote
1answer
46 views

Express $ G_y$ in terms of $G_x$. [duplicate]

A finite group $G$ acts on a finite set $X$, the action of $g \in G$ on $x \in X$ being denoted by $gx$. For each $x \in X$ the stabilizer of $x$ is the subgroup $G_x = \{g \in G : gx = x\}$. If $x, y ∈ ...
1
vote
2answers
69 views

Counterexample that $a\in G$, $a^n\notin H$, for $H$ a subgroup of finite index $n$ in $G$. [duplicate]

Let $G$ be a group and $H$ a subgroup of finite index $n$. Give a counterexample that $a\in G$, $a^n\notin H$ (although I can prove that there exists $k\in\{1,2,\dots,n\}$ such that $a^k\in H$). ...
-1
votes
2answers
118 views

Group acting on a set.

Let $G$ be a group of order $7$ acting on a set of $5$ elements. Show that the action of $G$ must have a fixed point.
0
votes
1answer
50 views

Question on equivariant functions and subconjugacy

I proved the following proposition as an exercise: Suppose $H \leq K \leq G$ are groups and that $G$ acts on $\frac{G}{H}$ and $\frac{G}{K}$. If $H$ is subconjugate to $K$ (i.e., if $\exists g \in ...
1
vote
2answers
113 views

Group Action Questions?

We discussed Group Actions in my undergraduate Modern Algebra class today. I understand the definition and example we went over in lecture, but the problem set is proving difficult. If I want to ...
2
votes
1answer
56 views

Relationship between decompositions of a $G$-variety $V$

Let $V$ be a variety over a field $k$, and let $G$ be an algebraic group over $k$ which acts morphically on $V$. $V$ has three canonical decompositions, and I'm interested in the relationships ...
5
votes
1answer
121 views

Two subgroups $H_1, H_2$ of a group $G$ are conjugate iff $G/H_1$ and $G/H_2$ are isomorphic

Let $H_1$ and $H_2$ be subgroups of some group $G$. Prove that the left $G$-sets $G/H_1$ and $G/H_2$ are isomorphic (as left $G$-sets) iff the subgroups $H_1$ and $H_2$ are conjugate. If $H_1$ ...
0
votes
2answers
396 views

About stabilizer in group action

Let $X$ be a finite set and $x$ is an element of $X$. Let $G_x$, the stabilizer subgroup, be the subset of $S_X$ consisting of permutations that fix $x$. The question is Is stabilizer always a ...
2
votes
1answer
74 views

When is the action of $G$ on $\text{Syl}_p(G)$ by conjugation is double transitive?

We know that the action of $G$ on $\text{Syl}_p(G)$ by conjugation is transitive. I wonder when this action can be double transitive on $\text{Syl}_p(G)$. Thanks for your help.
5
votes
3answers
329 views

Without using Sylow: Group of order 28 has a normal subgroup of order 7

Prove that a group of order 28 has a normal subgroup of order 7. How can I prove this without using Sylow's theorem? I know by Cauchy’s theorem, there exists an $x\in G$ with order 7, now I just ...
1
vote
3answers
93 views

Relationship between group actions and homomorphisms

I know that there exist no nontrivial homomorphism from $S_3$ into $Z_5$ as they are groups of co-prime order. I am not looking for an explanation of this but for an explanation concerning the obvious ...
7
votes
1answer
109 views

Understanding what an action is?

This is a very simple question, and I am quite embarrassed to ask it! I'm trying to understand what an action is in general, and perhaps the best place to start is to try and outline my current ...
1
vote
0answers
70 views

Lift a group action from a quotient

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 ...
0
votes
2answers
95 views

Orbit and Stabilizer

Are the following definitions essentially the same: Orbit: Let $G$ be a group of permutations of a set $S$. For each $s \in S$, let $\operatorname{orb}_G(s)= \{f(s) \mid f \in G\}$. The set ...
1
vote
1answer
56 views

Question about primitive group actions

In Glass' Partially Ordered Groups Corollary 7.4.4 says: If $G$ is an ordered group and $(G,G)$ is the right regular representation, then $(G,G)$ is primitive if and only if $G$ is ...
1
vote
2answers
111 views

Using counting formula to get |G| = |kernel φ||image φ|

The counting formula I am saying : Let S be a finite set on which a group G operates, and let Gs and Os be the stabilizer and orbit of an element s of S. Then |G|=|Gs||Os| or ...
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 ...
1
vote
1answer
90 views

Is the kernel of this group action the centralizer?

In Dummit and Foote, they state "... let the group $N_G(A)$ (normalizer) act on the set $A$ by conjugation. It is easy to check that the kernel of this action is the centralizer $C_G(A)$." From ...
5
votes
0answers
129 views

Why are they called orbits?

When we study actions in group theory, we consider sets of the form $$\text{Orb}_G(x)=\{gx\mid g\in G\} $$ that are called orbits. Although, the only reason I find convincing for that name is that in ...