Basics of Groups

Prove 1.11.

Let $i$ be a number such that $i^{2}=-1$. Let $\mathcal{S}=\{1,-1,i,-i\}$.

1. Find the identity of the group $(\mathcal{S},\times)$ where $\times$ denotes regular multiplication.

2. Find the orders of the elements in $(\mathcal{S},\times)$.

Find the other generator of the group $(\{1,-1,i,-i\},\times)$, where $i^{2}=-1$.

Simplify $rsr^{4}sr^{3}$ in the group $D_{6}$.

Prove that $\mathbb{Q}$ under addition forms an abelian group.(Note: addition is assumed to be associative and commutative by an exercise given elsewhere in the book and an exercise given elsewhere in the book respectively, so you do not need to prove them.)

Let $G$ be a group where every element in $G$ is its own inverse. Prove that $G$ is abelian.

Let $G$ be a group with identity $e$. Suppose an element $x\in G$ has finite order $n$. Prove that a positive integer $m$ is a multiple of $n$ if and only if $x^{m}=e$.(Hint: consider Euclid’s division lemma (an exercise given elsewhere in the book) to prove one direction of the claim.)

Let $G$ be a group.

1. Suppose $(gh)^{2}=g^{2}h^{2}$ for all elements $g,h\in G$. Prove that $G$ is abelian.

2. Suppose $G$ is abelian. Prove that $(gh)^{n}=g^{n}h^{n}$ for all elements $g$ and $h$ in $G$ and for all positive integers $n$.

Let the group $G=(\mathbb{Z}_{n},\oplus_{n})$. Show that $G$ is cyclic with order $n$.

Let the function $T:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be defined by

Define the set $A=\{T^{r}\mid r\in\mathbb{Z}\text{ and }r\geq 1\}$. Show that $A$ is a group under function composition ($\circ$) and state the order of this group.