Paper 3, Section II, D

Let $p$ be a prime number. Let $G$ be a group such that every non-identity element of $G$ has order $p$.

(i) Show that if $|G|$ is finite, then $|G|=p^{n}$ for some $n$. [You must prove any theorems that you use.]

(ii) Show that if $H \leqslant G$, and $x \notin H$, then $\langle x\rangle \cap H=\{1\}$.

Hence show that if $G$ is abelian, and $|G|$ is finite, then $G \simeq C_{p} \times \cdots \times C_{p}$.

(iii) Let $G$ be the set of all $3 \times 3$ matrices of the form

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

where $a, b, x \in \mathbb{F}_{p}$ and $\mathbb{F}_{p}$ is the field of integers modulo $p$. Show that every nonidentity element of $G$ has order $p$ if and only if $p>2$. [You may assume that $G$ is a subgroup of the group of all $3 \times 3$ invertible matrices.]

*Typos? Please submit corrections to this page on GitHub.*