Past exam of the groups and representations course of the University of Oxford / 2019 / 2 / a

Put $r=g_1=\mathrm{diag}(\alpha ,{\alpha }^2)$ and $s=g_2$, using ${\alpha }^{\ast }={\alpha }^{-1}={\alpha }^2$. ThenEvery word can therefore be put in the form $r^k$ or $r^{k}s$, with $k=0,1,2$. These give six distinct matrices:The group isomorphism $r\mapsto (123)$, $s\mapsto (12)$ identifies this group with the symmetric group $S_3$, with permutations composed from right to left.

Conjugation by $s$ exchanges $r,r^2$. Conjugation by $r$ cycles $s,rs,r^{2}s$, because $r(r^{k}s)r^{-1}=r^{k+2}s$. Their conjugacy classes arewith sizes $1,2,3$. There is no mixing between them: their elements have orders one, three and two, respectively.