Define the centre of a group, and prove that a group of prime-power order has a nontrivial centre. Show also that if the quotient group $G / Z(G)$ is cyclic, where $Z(G)$ is the centre of $G$, then it is trivial. Deduce that a non-abelian group of order $p^{3}$, where $p$ is prime, has centre of order $p$.

Let $F$ be the field of $p$ elements, and let $G$ be the group of $3 \times 3$ matrices over $F$ of the form

$$\left(\begin{array}{lll} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array}\right)$$

Identify the centre of $G$.