Group Actions

Let $G$ be a group. Let the function $\alpha:G\times G\to G$ be such that $\alpha(g,x)=gxg^{-1}$. Show that $\alpha$ is a group action of $G$ on $G$.

Let $X=\{1,2,3\}$, and let $\mathrm{S}_{3}$ act on $X$. What are the fixed point(s) of each of the 6 permutations in $\mathrm{S}_{3}$?

Let $X=\{1,2,3\}$, and let $\mathrm{S}_{3}$ act on $X$. What are the stabilizers of each of the 3 elements in $X$?

Let $G$ be a group that acts on a non-empty set $X$. Let $g,h\in G$ and $x\in X$. Prove that $g\cdot x=h\cdot x$ if and only if $g^{-1}h\in\mathrm{Stab}_{G}(x)$.

Let $G$ be a group that acts on a non-empty set $X$. Prove that

1. every orbit is non-empty;

2. every element in $X$ is in some orbit; and

3. if $\mathrm{Orb}_{G}(x_{1})\cap\mathrm{Orb}_{G}(x_{2})\neq\varnothing$ where $x_{1},x_{2}\in X$, then $\mathrm{Orb}_{G}(x_{1})=\mathrm{Orb}_{G}(x_{2})$.

Let $G$ be a group that acts on a non-empty set $X$. Show that if the group action is transitive if and only if $\mathrm{Orb}_{G}(x)=X$ for all $x\in X$.

Let $G=\mathrm{S}_{n}$ be a transformation group and consider the $G$-set $X=\{1,2,3,\dots,n\}$.

1. Show that the group action is transitive.

2. Find the order of the stabilizer of $x\in X$ by $G$.

Let $G$ be a group and $x\in G$. Prove that $|\mathrm{Cl}(x)|=[G:\mathrm{C}_{G}(x)]$.

The Cayley table of $D_{3}$ is in 1.29.

1. Find $\mathrm{Z}(D_{3})$, the center of $D_{3}$.

2. For each conjugacy class of $D_{3}$ with more than 1 element, find a representative.

3. Find the class equation of $D_{3}$.

For a finite group $G$ with order $np$ where $p$ is prime and $n$ is a positive integer, show that $G$ has a subgroup of order $p$.

Let $G=D_{5}$, the dihedral group of order 10.

1. What is/are the possible order(s) of a non-trivial proper subgroup of $G$?

2. For each of the possible order(s) identified in (i), find such a subgroup.

Suppose $G$ is a finite group with order $n>1$ where $g^{2}=e$ for all $g\in G$. Prove that $n=2^{k}$ where $k$ is a positive integer.

A group action is said to be free if $g\cdot x=x$ means that $g$ is the identity (i.e., only the identity fixes any $x$).

Let $G$ be a group and $S$ be a non-empty $G$-set. Suppose $G$ acts on $S$ freely and transitively. Prove that $G$ and $S$ have the same number of elements.

Let $G$ be a group of order 25 and $X$ be a $G$-set of 24 elements. Show that there exists an element in $G$ with a fixed point.

A bracelet consists of 3 beads that each can be one of $n$ colours. Two bracelets are the same if the rotation of one yields the other, or if one can be obtained via reflecting about a line, or any combination of these two actions. How many distinct bracelets are there?

Let $p$ be a prime number. Prove that a group of order $p^{2}$ must be abelian.(Hint: Consider 1.3 and 1.11.)

Let $G$ be a finite $p$-group, and $X$ be a $G$-set. Let the set

Prove that $|X|\equiv|\Omega|\pmod{p}$.