(a) Let $x$ be an element of a finite group $G$. Define the order of $x$ and the order of $G$. State and prove Lagrange's theorem. Deduce that the order of $x$ divides the order of $G$.

(b) If $G$ is a group of order $n$, and $d$ is a divisor of $n$ where $d<n$, is it always true that $G$ must contain an element of order $d$ ? Justify your answer.

(c) Denote the cyclic group of order $m$ by $C_{m}$.

(i) Prove that if $m$ and $n$ are coprime then the direct product $C_{m} \times C_{n}$ is cyclic.

(ii) Show that if a finite group $G$ has all non-identity elements of order 2 , then $G$ is isomorphic to $C_{2} \times \cdots \times C_{2}$. [The direct product theorem may be used without proof.]

(d) Let $G$ be a finite group and $H$ a subgroup of $G$.

(i) Let $x$ be an element of order $d$ in $G$. If $r$ is the least positive integer such that $x^{r} \in H$, show that $r$ divides $d$.

(ii) Suppose further that $H$ has index $n$. If $x \in G$, show that $x^{k} \in H$ for some $k$ such that $0<k \leqslant n$. Is it always the case that the least positive such $k$ is a factor of $n$ ? Justify your answer.