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.
10
votes
2answers
100 views
What's the idea of an action of a group?
I know the formal definition of an action over a set. I'm not asking this.
What I'm asking is: what's the intuition of it? It is a way to define an algebra over a set? Since an action can exist in ...
10
votes
0answers
72 views
Show that $h \equiv 1 \pmod p$, where $h$ is the number of subgroups of order $p$ and $p$ divides the group order. [duplicate]
Let $G$ be a finite group and $p$ a prime number that divides the
order of $G$. Let $h$ be the number of subgroups of $G$ of order $p$.
Prove that there are $h(p-1)$ elements of order $p$ in ...
8
votes
3answers
203 views
What does an outer automorphism look like?
I am working on a project in my group theory class to find an outer automorphism of $S_6$, which has already been addressed at length on this site and others. I have a prescription for how to go about ...
6
votes
1answer
103 views
Group actions transitive on certain subsets
Let $G$ be a group acting on a finite set $X$. This also gives an action of $G$ on the subsets of $X$ of any given size, and we can ask whether this action is transitive for some specified size of ...
6
votes
1answer
77 views
Natural way to define a free action of a finite abelian group
Let $G$ be a finite abelian group. Then $G \simeq \mathbb{Z}_{u_1} \oplus \cdots \oplus \mathbb{Z}_{u_m}$, where $u_{i}$ is a power of some prime number. Without loss of generality I will consider $G ...
6
votes
1answer
86 views
Invariants of binary forms under a $\begin{pmatrix} 1& 1 \\ 0& 1 \end{pmatrix}$ action
The special linear group $\text{SL}_2(\mathbb{Z})$ of $2\times 2$ invertible matrices in $\mathbb{Z}$ acts on binary cubic forms $\{ax^3 + bx^2y + cxy^2 + dy^3\}$ by acting on the vector $(x,y)^T$. ...
6
votes
0answers
34 views
Chern classes of free quotient manoflds
Let $X$ be a compact complex manifold. Assume that a finite group acts on $X$ freely. Then the quotient $X/G$ is again a compact complex manifold. I wonder if there is a good way to compute Chern ...
5
votes
3answers
108 views
Group Action - Permutation on the Polynomial
I'm trying to check the permutation on the polynomial is a Group Action, but I'm not getting the second axiom. I'm following my lecturer's work --- Examples 2.1 and 2.6 on page 5 on ...
5
votes
2answers
245 views
Kernel of Group Action
this is my first post here.
I have a question regarding a proof in Algebra by Hungerford:
Let $G$ be a group and $H$ a subgroup of $G$. Let $S$ be the set of all cosets of $H$, where $G$ acts on ...
5
votes
1answer
74 views
A question about quotient under group action
Let $X$ be a Hausdorff space, and $G$ a group acting on $X$ by homeomorphisms. Let $H$ be a normal subgroup of $G$. Is it true that $X/G$ is homeomorphic to $(X/H)/(G/H)$ ?
If so, can you please ...
5
votes
1answer
48 views
How to check the strong ergodicity of the $SL_2(\mathbb{Z})$-action on the torus?
Suppose $\Gamma\subset SL_2(\mathbb{Z})$ is a non-amenable subgroup, especially, $\Gamma=SL_2(\mathbb{Z})$. Consider the natural action of $\Gamma$ on $S^1\times S^1=T^2$.
How to check that this ...
4
votes
3answers
84 views
Do we gain anything interesting if the stabilizer subgroup of a point is normal?
Let $G$ be a group and $S$ a $G$-set with action $(g,s) \mapsto gs$. For some $s \in S$, let the stabilizer of $s$, $G_s=\{g \in G\,|\,gs=s\}$ be normal in $G$. What does this let us say about the ...
4
votes
2answers
135 views
If $\exp(itH) A \exp(-itH) = A$ for all $t$, do $A$ and $H$ commute?
Let $H$ be a self-adjoint $n \times n$ matrix with complex entries. $H$ gives rise to a continuous 1-parameter group of unitaries $t \mapsto U_t = \exp(itH) : \mathbb{R} \to U(n)$.
Let $A$ be some ...
4
votes
3answers
53 views
at least one element fixed by all the group
$G$ is a p-group and $S$ is a set that $G$ acts on. p does not divide $|S|$. Why is there at least one element $a\in S$ such that $|O(a)|=1$, or in other words, $G_a=G$?
I tried to ask this question ...
4
votes
2answers
70 views
About the category $\mathrm{Set}(G)$
I'm not good with categories. I've attempted several times to understand what a natural transformation is, and so far I've failed. But I'm trying to learn algebraic topology now, and it seems that I ...
4
votes
1answer
98 views
wiki's definition of “strongly continuous group action” wrong?
Wikipedia defines strongly continuous group action as follows:
A group action of a topological group G on a topological space X is said to be
strongly continuous if for all x in X, the map g ↦ ...
4
votes
1answer
93 views
Continuous Actions and Homomorphisms
I am learning about the compact-open topology and have a small proposition I am struggling to prove. Let $G$ be a topological group, $X$ a compact, Hausdorff space, and $H(X)$, the homeomorphisms of ...
4
votes
2answers
71 views
Free objects in $\mathrm{Set}(G).$
What are the free objects in the category of $G$-sets for a group $G$?
After considerable deliberation (I'm not very bright), I'm pretty sure they are the $G$-sets $X$ on which $G$ acts freely, that ...
4
votes
1answer
97 views
Orbits of $\mathbb{Z}/n\mathbb{Z}$ under automorphism subgroup action
Let $A\leq \rm{Aut}(\mathbb{Z}/n\mathbb{Z})$. I am interested in the orbits of $\mathbb{Z}/n\mathbb{Z}$ under the action of $A$, i.e. the sets
$$\{\sigma i : \sigma \in A\},$$
where $i\in ...
4
votes
0answers
62 views
The classifying space of a gauge group
Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by
$$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = ...
4
votes
0answers
42 views
Equiv Relation of Orbits - Group Action [duplicate]
Let $G$ be a group that acts on $X$. I want to show that the orbits of $G$ partition $X$. I am given the relation $x\sim y \iff x\in Orb(y)$. Now:
$x\sim y\iff x\in Orb(y) \iff x=gy$ for some $g\in G ...
4
votes
1answer
165 views
Quotient of a locally compact Hausdorff space by a proper action is Hausdorff
I am trying to prove the following:
Let $G$ be a topological group acting properly on a Hausdorff locally
compact space $X$, i.e. preimages of compacts sets by the map
$$G\times X\to X\times ...
4
votes
0answers
147 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 ...
3
votes
3answers
110 views
Examples of the dihedral group $D_4$ acting on sets
Consider the group $D_4$. Give examples of $D_4$ acting on a set.
Attempt: So $|D_4| = 8$. I have come up with a few, but I was wondering what some people here thought.
First one we came up with ...
3
votes
1answer
114 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
2answers
100 views
Original papers on the subject of group actions
Does anyone if there are any original paper(s) that first introduced the notion of group action or permutation representation, and who the author(s) were? Any references I have found so far on e.g. ...
3
votes
1answer
34 views
Infinite imprimitive non abelian group?
My new question is
Is there an infinite, imprimitive and non abelian group?
Thank you for the further answers.
3
votes
1answer
57 views
Is the Structure Group of a Fibre Bundle Well-Defined?
Am I right in thinking that the structure group of a fibre bundle is any group $G$ of homeomorphisms of the fibre $F$ such that all transition functions map into $G$? Or is $G$ somehow the minimal ...
3
votes
1answer
63 views
The order of a conjugacy class is bounded by the index of the center
If the center of a group $G$ is of index $n$, prove that every conjugacy class has at most $n$ elements. (This question is from Dummit and Foote, page 130, 3rd edition.)
Here is my attempt: we have
...
3
votes
1answer
133 views
Elliptic Points of Modular Group in Upper Half Plane
This is a very small question.
Let $\mathbb{\Gamma} = \mathrm{SL_2}(\mathbb{Z})$ be the modular group, $\mathcal{F} = \{z \in \mathbb{C} ;\; \lvert z \rvert \geq 1,\; \lvert \Re (z) \rvert \leq ...
3
votes
2answers
223 views
Groups acting on topological spaces
In algebraic topology, we show that a group $G$ acting properly discontinously on a simply connected (and sufficently connected) topological space $Y$ is isomorphic to the fundamental group of $Y/G$.
...
3
votes
1answer
25 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 : ...
3
votes
2answers
150 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 ...
3
votes
1answer
55 views
Conjugation on subgroups of $A_4$ faithful?
Let $X$ be the set of all subgroups of $G=A_4$. We define the group action $$G\times X\ni(g,H)\mapsto gHg^{-1}\in X$$
I am trying to determine whether this action is faithful, i.e. $\bigcap_{H\in X} ...
3
votes
1answer
117 views
Circle Acting on Circle/Ball
Is much known about $S^1$-actions on the following simple spaces?:
1) $D^2$ the disk
2) More generally $D^n$ the n-ball
3) $S^1$ the circle
In particular, does every $S^1$-action on the disk (or ...
3
votes
0answers
36 views
burnside lemma cube [duplicate]
Having n colors, use the lemma to find a formula for the number of ways to color the edges of the cube.
What I have so far:
I got $|A/G| = \dfrac{n^{12} + 6n^3 + 3n^6 + 8n^4 + 6n^7}{24}$ but when I ...
2
votes
3answers
150 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 ...
2
votes
2answers
241 views
Complex projective line hausdorff as quotient space
I was wondering if there is a simple argument showing that the complex projective line defined as $\mathbb{CP^1} = \big(\mathbb{C}^2 \setminus \{0\}\big)/{\mathbb{C}^{\times}}$ is hausdorff when ...
2
votes
2answers
109 views
Parabolic isometries on Gromov hyperbolic spaces
Let $X$ be a $\delta$-hyperbolic geodesic space. Then we have the following classification of isometries on $X$:
Theorem: Let $g$ be an isometry on $X$. Then, exactely one of the following case ...
2
votes
1answer
60 views
Confused over group action
Say I want to put action of $S_4$ on $V^{\otimes 4}$. If I want to act on the left, why can't I say
$$\sigma(v_1 \otimes v_2 \otimes v_3 \otimes v_4) = v_{\sigma(1)} \otimes \ldots \otimes ...
2
votes
1answer
57 views
What does “lifted action” mean?
I read about angular moment and linear moment but I don't know what "lifted action" means. Can you explain please?
Thanks. :)
2
votes
1answer
69 views
Motivation for the term “transitive” group action
I have two questions:
In a text, I read that a group permutes pairs of faces of a solid transitively. Geometrically, what are they referring to, and what is an example of when a group may not ...
2
votes
2answers
195 views
Algebra - Infinite Dihedral Group
Let $G$ be the set of bijections $\mathbb{R} \to \mathbb{R}$ which preserve the distance between pairs of points, and send integers to integers. Then $G$ is a group under composition of functions. The ...
2
votes
1answer
99 views
Extending a group action to a quotient group
If I have a cyclic group $G = (a)$ acting on an abelian group $A$, I need to define a natural action of $G$ on the quotient space $A/B$, where $B$ is a normal subgroup of $A$ with the property that ...
2
votes
2answers
37 views
Homsets of group actions related to fixed points
MacLane and Moerdijk's Sheaves in Geometry and Logic has a section on Continuous Group Actions (Sec. III.9). On page 152, there is an isomorphism displayed:
$$Hom_G(G/U, X) \cong X^U$$
In their ...
2
votes
2answers
70 views
How to show $\mathbb R^n/\mathbb Z^n$ is diffeomorphic to torus $\mathbb T^n$?
Suppose the additive group $\mathbb Z^n$ acts on $\mathbb R^n$ through translation. How to show $\mathbb R^n/\mathbb Z^n$ is diffeomorphic to torus $\mathbb T^n$? The translation action is given by ...
2
votes
1answer
110 views
Using Burnside's lemma on the cube.
Having $n$ colors, use the lemma to find a formula for the number of ways to color the edges of the cube.
Here is what I have so far: The Burnside lemma says that $\displaystyle |X/G| = ...
2
votes
1answer
82 views
In a group of Möbius transformations, does discontinuity imply discreteness?
Let $G$ be a subgroup of the group of Möbius transformations
$$ z \mapsto \frac{az+b}{cz+d}.$$
What is the relationship between the two conditions:
(1) $G$ being discrete.
(2) $G$ acting properly ...
2
votes
1answer
48 views
Virtually infinite cyclic groups act on a tree
A virtually infinite cyclic group $G$ is quasi-isometric to $\mathbb{Z}$ and thus has two ends; by Stallings theorem, $G$ acts (without inversion) on a tree with finite edge-stabilizers.
But the ...
2
votes
2answers
48 views
Rotman's Introduction to to the theory of groups. Exercise 3.45.
Can you give me a hint on the first part of the exercise?
Let $p$ be a prime and let $X$ be a finite $G$-set, where $|G| = p^n$ and $|X|$ is not divisible by $p$. Prove that there exists $x \in X$ ...
