State and prove Lagrange's Theorem.

Hence show that if $G$ is a finite group and $g \in G$ then the order of $g$ divides the order of $G$.

How many elements are there of order 3 in the following groups? Justify your answers.

(a) $C_{3} \times C_{9}$, where $C_{n}$ denotes the cyclic group of order $n$.

(b) $D_{2 n}$ the dihedral group of order $2 n$.

(c) $S_{7}$ the symmetric group of degree 7 .

(d) $A_{7}$ the alternating group of degree 7 .