| course |
course_year |
question_number |
tags |
title |
year |
Groups, Rings and Modules |
IB |
35 |
IB |
2009 |
Groups, Rings and Modules |
|
Paper 2, Section II, F |
2009 |
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 $G$.