Paper 3, Section II, D

Groups | Part IA, 2014

Let pp be a prime number. Let GG be a group such that every non-identity element of GG has order pp.

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

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

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

(iii) Let GG be the set of all 3×33 \times 3 matrices of the form

(1ax01b001)\left(\begin{array}{lll} 1 & a & x \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}\right)

where a,b,xFpa, b, x \in \mathbb{F}_{p} and Fp\mathbb{F}_{p} is the field of integers modulo pp. Show that every nonidentity element of GG has order pp if and only if p>2p>2. [You may assume that GG is a subgroup of the group of all 3×33 \times 3 invertible matrices.]

Typos? Please submit corrections to this page on GitHub.