Let $R$ be a relation on the set $S$. What does it mean for $R$ to be an equivalence relation on $S$ ? Show that if $R$ is an equivalence relation on $S$, the set of equivalence classes forms a partition of $S$.

Let $G$ be a group, and let $H$ be a subgroup of $G$. Define a relation $R$ on $G$ by $a R b$ if $a^{-1} b \in H$. Show that $R$ is an equivalence relation on $G$, and that the equivalence classes are precisely the left cosets $g H$ of $H$ in $G$. Find a bijection from $H$ to any other coset $g H$. Deduce that if $G$ is finite then the order of $H$ divides the order of $G$.

Let $g$ be an element of the finite group $G$. The order $o(g)$ of $g$ is the least positive integer $n$ for which $g^{n}=1$, the identity of $G$. If $o(g)=n$, then $G$ has a subgroup of order $n$; deduce that $g^{|G|}=1$ for all $g \in G$.

Let $m$ be a natural number. Show that the set of integers in $\{1,2, \ldots, m\}$ which are prime to $m$ is a group under multiplication modulo $m$. [You may use any properties of multiplication and divisibility of integers without proof, provided you state them clearly.]

Deduce that if $a$ is any integer prime to $m$ then $a^{\phi(m)} \equiv 1(\bmod \mathrm{m})$, where $\phi$ is the Euler totient function.