Paper 3, Section II, D

(a) Let $G$ be a non-trivial group and let $Z(G)=\{h \in G: g h=h g$ for all $g \in G\}$. Show that $Z(G)$ is a normal subgroup of $G$. If the order of $G$ is a power of a prime, show that $Z(G)$ is non-trivial.

(b) The Heisenberg group $H$ is the set of all $3 \times 3$ matrices of the form

$\left(\begin{array}{lll} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{array}\right)$

with $x, y, z \in \mathbb{R}$. Show that $H$ is a subgroup of the group of non-singular real matrices under matrix multiplication.

Find $Z(H)$ and show that $H / Z(H)$ is isomorphic to $\mathbb{R}^{2}$ under vector addition.

(c) For $p$ prime, the modular Heisenberg group $H_{p}$ is defined as in (b), except that $x, y$ and $z$ now lie in the field of $p$ elements. Write down $\left|H_{p}\right|$. Find both $Z\left(H_{p}\right)$ and $H_{p} / Z\left(H_{p}\right)$ in terms of generators and relations.

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