Tagged Questions
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 $x \in X$ ...
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 ...
1
vote
1answer
32 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
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
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} ...
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 ...
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
1answer
109 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| = ...
0
votes
1answer
80 views
p-group and group actions
$G$ is a $p$-group, which means $|G|=p^n$ for $n\in \mathbb{Z^+}$.
Now,if $p$ does not divide $|S|$, for S is a set that G acts upon, how do I show that there exists $a\in S$ such that $G_a=G$
So ...
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 ...
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. :)
0
votes
2answers
53 views
Group and orbit question.
Suppose group $G$ acts on a set $A$.
a) If $x$ and $y$ are in the same orbit, show that there exists some $g \in G$ such that $gG_x g^{-1} = G_y$.
b) Show that if $|G.x|$ is finite, then $|G.x| = ...
0
votes
1answer
68 views
Orbit and stabilizer question.
Let $K$ be a field. Consider the action of the multiplicative group $K^* := K-\{0 \}$ on the vector space $K^n$ given by scaling.
a) Describe the orbits of this action.
b) Describe the stabilizer ...
-3
votes
2answers
127 views
On Conjugacy Classes and Alternating Group $A_n$
in Dummit & Foote
in page 131
"
Let $K$ be a conjugacy class and suppose that $K$ is subset of $A_n$ .
1.Show that if $\sigma$ belongs to $S_n$ then , $\sigma$ does not commute with any odd ...
0
votes
0answers
60 views
Actions of G on corresponding orbits are equivalent for stable maps
Given actions of G on X and on Y, these actions are equivalent if and only if there is a bijection from $X $ \ $ G \rightarrow Y$ \ $G$ so that actions of G on corresponding orbits are equivalent.
...
1
vote
2answers
66 views
Finding the number of orbits
How many orbits are there of $(12)(25)$ in $S_{5}$?
Considering the permutation $(12)$, it has $4$ orbits and is as follows:
$\{\{1,2\},\{3\},\{4\},\{5\}\}$
and (25) also has 4 orbits and is also ...
3
votes
1answer
62 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
...
1
vote
2answers
92 views
Properties of set $\mathrm {orb} (x)$
Properties of set $\mathrm {orb} (x)$:
${\displaystyle \bigcup_{x\in X}\mathrm{orb}(x)=X}$;
$\mathrm{orb}(x)\cap\mathrm{orb}(y)=\emptyset$
for all $x,y\in X, x\neq y$
How to prove it? Please ...
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 $G$
...
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 ...
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. ...
0
votes
2answers
66 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 ...
8
votes
3answers
202 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 ...
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 ...
