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.

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4
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1answer
117 views

Given a finite non abelian group G, prove the order of the center is less than or equal to 1/4 of the order of the group.

I am a bit confused on where to start with this one. The question given was Let $G$ be a non abelian finite group. Prove $|Z(G)| \leq \frac {1}{4} |G|$. Am I supposed to use some property of ...
1
vote
0answers
35 views

Define a group $G=<x,y:o(x)=4,o(y)=2,o(xy)=3>$. show that $G \cong S_{4}$

I know that $S_{4}=<(1,2,3,4),(1,2)>$, but don't know how to prove the isomorphism
1
vote
0answers
27 views

The set $\{ x \in G : \alpha \in \Sigma^x \}$ for a minimal base $\Sigma$ of permutation group

Let $G$ a group acting on $\Omega$. A subset $\Sigma \subseteq \Omega$ is called a base if $$G_{(\Sigma)} := \{ x \in G : \delta^x = \delta \mbox{ for all } \delta \in \Sigma \} = 1$$ (i.e. the ...
1
vote
0answers
25 views

Shortcuts when computing with permutations or other group related constructs

In some kind well-known is that for cycles (and therefore each permutation written as a product of disjoint cycles) the conjugation by another permutation could be easily computed, just replace the ...
0
votes
0answers
14 views

Verification about group actions and “uniformity” of an action

I spent some time revisiting group actions this week. I was hoping to get someone to verify a seemingly straightforward claim. I also had a thought on how "uniform" an action is on a space. ...
2
votes
0answers
80 views

Proof that $A_n$ is simple for $n \ge 5$, is the one presented here overcomplicated?

In the book Permutation Groups by Dixon & Mortimer, page 78, the well-known fact that for $|\Omega| \ge 5$ the alternating group $Alt(\Omega)$ is simple is proven. It uses a Theorem that if a ...
0
votes
1answer
58 views

A special subgroup of $S_{10}$ generated by four involutions

Is there an easy way to describe the subgroup of $S_{10}$ generated by the four involutions: \begin{align*} & (4 ~ 7)(5 ~ 8)(6 ~ 9) \\ & (2 ~ 7)(3 ~ 8)(6 ~ 10) \\ & (1 ~ 7)(3 ~ 9)(5 ~ ...
0
votes
2answers
42 views

Number of Involutions generating Symmetric Group

How to show that strictly less than $n-1$ involutions (of which the transpositions are a special case) could not generate $S_{n}$ for $n > 3$? I know that $n-1$ transpositions are sufficient, or ...
0
votes
0answers
40 views

Order of elements in group and in permutation representation of that group

If $G \le Sym(\Omega)$ and for some subset $\Gamma \subseteq \Omega$ we have $\Gamma^x = \Gamma$ for each $x \in G$ (such a set is called $G$-invariant) then each element $x \in G$ could be seen as a ...
0
votes
0answers
27 views

About relation between simple section and section in induced representation on orbits

The following is from Dixon & Mortimer, Permutation Groups, page 74. A group $S$ is called a section of a group $G$ if for some subgroups $H$ and $K$ of $G$ we have $K \unlhd H$ and $H/K \cong ...
0
votes
1answer
42 views

$\mathbb{Z}^2$ acts on $\mathbb{R}^2$ by translation is 'separable'

I have to study the following G-Action $$ \begin{cases}\mathbb{Z}^2 \times \mathbb{R}^2 & \longrightarrow \mathbb{R}^2 \\ (m,n)\times (x,y) & \longmapsto (x+m,y+n) \end{cases} $$ That is, as ...
0
votes
0answers
16 views

Question on Proof that in a primitive group we have for the subdegrees $n_{i+1} \le n_i(n_2-1)$

The following question is on a proof from Dixon & Mortimer, Permutation groups, page 72. He uses the characterisation that a transitive group $G$ acts primitive on $\Omega$ iff for each orbital ...
0
votes
0answers
24 views

Stabilizer over G acting on itself by conjugation

For elements $g_i, g_j \in G$, the stabilizer of $g_j$ over this group action is the set $\{g_i \in G: g_i g_j g_i^{-1} = g_j\}$. Now, since $g_i g_j g_i^{-1} = g_j \iff g_i = g_j g_i g_j^{-1}$, would ...
0
votes
1answer
25 views

$S_n$ acting on set of natural numbers

When it's said that the group of permutations $S_n$ acts on $A=\{1,...,n\}$, how does an element $\sigma_i \in S_n$ act on a single element $a \in A$? In other word, how does a permutation of numbers ...
0
votes
1answer
42 views

Roundabout way to prove the commutator subgroup is normal

I think I have somewhat of a roundabout way to show that the commutator subgroup $[G,G]$ is normal in $G$. Please check my proof for errors/improvements. Let $G$ be a group and define $S := ...
0
votes
0answers
21 views

If subdegree $\ge n/4$ and orbital not self-paired, then the orbital graph is connected with paths of length $\le 2$

Let $G$ be a finite transitive permutation group of degree $n$, the list of subdegrees (the lengths of the orbits of one of the point stabilizers) is an invariant of $G$. We shall denote them in ...
2
votes
1answer
30 views

How to prove this action is discontinuous?

Let $G$ be a group of homeomorphisms of a topological space $X$. The action of $G$ on $X$ is said to be discontinuous at a point $x \in X$ if $G_x :=$ the stabilizer of $x$, is finite. $x$ has an ...
2
votes
2answers
22 views

Why do we need the action to be effective?

Let $G$ be a group of homeomorphisms of a connected, locally path connected, locally compact metric space $X$. Let Homeo$(X)$ be the group of homeomorphisms of $X$ endowed with the compact open ...
0
votes
1answer
36 views

Missing crucial steps in Fischer Algebra proof on class equation of groups

I have the following Proposition in Fischer's Algebra Proposition: Let $G$ be a group and $\alpha: G \times M \to M$ a $G$-Action on the finite set $M$. Let $a_1, \dots , a_r$ be a representatives ...
3
votes
1answer
28 views

Is there an relation between the notions of continuous and discontinuous group actions?

Let $G$ be a group of homeomorphisms of a topological space $X$. The action of $G$ on $X$ is said to be discontinuous at a point $x \in X$ if $G_x :=$ the stabilizer of $x$, is finite. $x$ has an ...
2
votes
0answers
40 views

Group acts strongly primitive iff point stabilizer is maximal subsemigroup

If a group $G$ acts on some set $\Omega$, then we have a natural action on $\Omega\times\Omega$ given by $(\alpha,\beta)^x := (\alpha^x, \beta^x)$. If $\Delta \subseteq \Omega\times\Omega$ is some ...
1
vote
0answers
29 views

Relation between orbitals and cosets of point stabilizer

Let $G$ be a group acting on a set $\Omega$. Then there exists a natural action of $G$ on $\Omega \times \Omega$ given by $(\alpha, \beta)^x = (\alpha^x, \beta^x)$. The orbits on $\Omega\times \Omega$ ...
3
votes
0answers
44 views

Verify proof to show this is a $\sigma$ - locally finite basis.

Can someone tell me if what I have done is correct? Proposition : Let $X$ be a compact metric space and $G$ a finite group acting on it. Let $p : X \rightarrow X/G$ be the orbit map. Let $B = \{ ...
5
votes
1answer
52 views

Show that $ \frac {\mathbb Z}{p \mathbb Z} \times \frac {\mathbb Z}{p \mathbb Z}$ is not isomorphic to Aut($G$) for any abelian group $G$

Here is a problem on which I'm stuck: Show that $ \frac {\mathbb Z}{p \mathbb Z} \times \frac {\mathbb Z}{p \mathbb Z}$ is not isomorphic to Aut($G$) for any abelian group $G$. On the Contrary, ...
2
votes
1answer
51 views

Show that if $p$ is a prime number, and $G$ is a transitive subgroup of $S_p$, then $G$ must contain a cycle of length $p$.

Note that a subgroup $G$ of $S_n$ is called a transitive subgroup of $S_n$ if it acts transitively on the set $\{1,2,...n\}$. I understand that, for any $x$, $y$ $\in \{1,2,...p\}$, there exists an ...
1
vote
1answer
33 views

Show that $G/N$ acts faithfully on $S$ if and only if $N=\ker\phi$

There is this supposed to be a not-so-difficult proof but somehow I just find it a bit hard to connect the dots. Suppose that $G$ acts on a set $S$, and let $\phi$ be the associated homomorphism from ...
0
votes
1answer
127 views

Show that elements are not conjugate to their inverse in groups of odd order.

Show that if G is a group of odd order, then no $x\in G$ other than the identity is conjugate to its inverse. We can't have elements of order 2, since by Lagrange theorem this would mean we ...
4
votes
4answers
33 views

Example of a group action G on a vector space V that fails to be linear (i.e. fails to be a linear representation).

I have seen the following definition of a linear group representation from C. Lent's notes on Representation Theory: A linear representation $ρ$ of $G$ on a complex vector space $V$ is a ...
4
votes
0answers
56 views

Elementary consequences of commuting limits and colimits over groups

In this n-cat cafe post, it is proven that for finite groups $G,H$ of coprime order, $G$-colimits and $H$-limits commute. Later on the following theorem is mentioned: Theorem 1. $H$-limits commute ...
0
votes
1answer
44 views

The general linear group over the complex numbers conjugate some proper subgroup

Let $G = Gl_{2}(\mathbb{C})$ and let $H = \left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} \ |\ a,b,c \in \mathbb{C}, ac \neq 0 \right\}$. Prove that every element of G is conjugate to ...
6
votes
1answer
80 views

Embedding of symmetric groups into orthogonal groups

Let $\Sigma_n$ be the symmetric group of order $n$. Let $O(N)$ be the orthogonal group acting on $\mathbb{R}^N$. Question: Given a fixed $N$, how to find the maximal $n$ such that $\Sigma_n$ can be ...
0
votes
1answer
26 views

Group acting on set A by its generators

If I act using right or left action by generators of some group G on some set A will I get a subgroup of $S_A$ isomorphic to G ? If so can someone provide me with a reason. For example: Use the ...
2
votes
2answers
47 views

The affine special linear group acts doubly transitive

Let $F$ be a field and $d \ge 2$. Denote by $ASL_d(F)$ the affine special linear group, i.e. the group of all transformation on $F^d$ with $t_{A,v}(u) = Au + v$ and $\det A = 1$. I want to show that ...
1
vote
1answer
14 views

Relation between general linear groups for subfield $K$ of $F$ of finite index.

If $F$ is a field and $d \ge 1$. Let $K$ be a subfield of $F$ with finite index $k = [F : K]$. Then $F$ is a $k$-dimensional vector space over $K$. Thus every $F$-vector space is also a $K$-vector ...
1
vote
1answer
66 views

stabilizer and kernel

Let $G$ be a permutation group on the set $A$, let $\sigma \in G$ and let $a \in A$. Prove that $\sigma G_A\sigma^{-1} = G_{\sigma(a)}$. Deduce that if $G$ acts transitively on $A$ then ...
3
votes
1answer
24 views

The wreath product $K ~\mbox{wr}_{\Gamma} ~ H$ acts faithful on $\Delta \times \Gamma$ iff $K$ acts faithful on $\Delta$

Let $K$ and $H$ be groups, and let $H$ act on $\Gamma$. Also let $\operatorname{Fun}(\Gamma, K) = \{ f : f : \Gamma \to K \}$ be the set of all functions from $\Gamma$ to $K$. This set is a group ...
2
votes
1answer
43 views

What does it mean for a subgroup to fix a vector space?

I was wondering if anyone could explain what is meant by the two statements below? Any help would be greatly appreciated. Let $U=\mathbb{F}_{q}^{m}$ and $W=\mathbb{F}_{q}^{n}$ be two vector spaces of ...
0
votes
1answer
39 views

Yoneda Lemma in context of a group action

On the representable functors page on Wikipedia, they mention that, if we consider a group $G$ as a category with a single object $\star$ and all arrows invertible, we can view a $G$-action on a set ...
1
vote
1answer
20 views

Cycle decomposition of given action

I don't understand the second part of this question at all could somebody explain to me what they want me to do for the second part? Let $S_3$ act on the set $\Omega$ of ordered pairs: $\{(i,j) : 1 ...
5
votes
2answers
67 views

Is the following scenario possible for stabilizer

So I was solving couple of problems involving stabilizer of group actions. I was wondering can we have kernel of group action being equal to the identity while all the stabilizers of G not being equal ...
2
votes
1answer
59 views

Prove that the map $H → O$ defined by $h → hx$ is a bijection , use this result to deduce Lagrange's Theorem

Exercise: Let $H$ be a subgroup of the finite group $G$ and let $H$ act on $G$ (here $A = G$) by left multiplication. Let $x \in G$ and let $O$ be the orbit of $ x$ under the action $H$. Prove ...
1
vote
1answer
85 views

Prove the relation ~ on $A$ defined by $a$~$b$ if and only if $a = hb$ is an equivalence relation.

Let $H$ be a group acting of a set $A$. Prove that the relation ~ on $A$ defined by $a$~$b$ if and only if $a = hb$ for some $h \in H$ is an equivalence relation. (For each $x\in A$ the equivalence ...
1
vote
1answer
25 views

If $G<S_n$ is transitive, calculate $1/|G| \cdot \sum_{g \in G} f(G)$

$G<S_n$ is transitive calculate $1/|G| * \sum_{g \in G} f(g)$ where $G<S_n$ and $f(g) = |\{ 1 \le i \le n | g(i) = i \}|$ I tried to use the orbit stabiliser theorem but didn't get anywhere ...
1
vote
3answers
49 views

If $G$ acts on $\Omega$ and $|G| = |\Omega| + 1$, show there exists nontrivial element fixing point without Burnside's lemma

Let $G$ act on $\Omega$ transitively, and let $|G| = |\Omega| + 1$ (both sets are assumed to be finite). I want to show from first principles (using maybe arguments like the pigeonhole principle, but ...
1
vote
2answers
39 views

The action of the group $SL_3(\mathbb{F}_p)$ on the group $\mathbb{F}_p^3/\{0\}$

Proof that the natural action of $SL_3(\mathbb{F}_p)$ on the group $\mathbb{F}_p^3/\{0\}$ is transitive action. ($\mathbb{F}_p$ is finite field of order p which is prime) Any clue?
4
votes
2answers
84 views

Does orbit-stabilizer theorem holds for monoid action?

For a group $G$ acting on some space $X$ we know there is a orbit-stabilizer theorem. My question is does this formula holds for monoid action? I think this formula may not hold, as inverse do not ...
2
votes
2answers
91 views

Normal subgroups of transitive group actions are primitive

If $G$ acts transitive on $\Omega$, a subset $\Delta \subseteq \Omega$ is called a block if for each $x \in G$ we have either $\Delta^x = \Delta$ or $\Delta^x \cap \Delta = \emptyset$. The singletons ...
4
votes
0answers
29 views

If $G$ acts primitively and $\Gamma \subseteq \Omega$ is not a block, then each pair of points could be separated

Let $G$ act on $\Omega$. A subset $\Delta \subseteq \Omega$ is called a block if for each $x \in G$ either $\Delta^x \cap \Delta = \emptyset$ or $\Delta^x = \Delta$, where $\Delta^x := \{ \delta^x : ...
4
votes
1answer
45 views

If $G$ acts transitively and $\Gamma \subseteq \Omega$ is not a block, then each pair of points could be separated

Let $G$ act transitively on $\Omega$. A subset $\Delta \subseteq \Omega$ is called a block if for each $x \in G$ either $\Delta^x \cap \Delta = \emptyset$ or $\Delta^x = \Delta$. If $\Gamma \subseteq ...
2
votes
1answer
44 views

Let $G$ act transitive, and $H \unlhd G$. Why is number of orbits at most $|G : H|$?

I have a question on the argument in part (2) of the following theorem (it is a shortened version of this theorem from the book Permutation Groups by Dixon & Mortimer): Theorem: Let $G$ be a ...