Past exam of the groups and representations course of the University of Oxford / 2019 / 1 / c

Write $A=g_1$, $B=g_2$ and $C=AB$. Direct multiplication givesThe relation $BA=-AB$ moves all occurrences of $A$ to the left of all occurrences of $B$. Removing even powers reduces every product of the generators to $\pm A^a{B}^b$ with $a,b\in \{0,1\}$. The resulting eight matrices are distinct, soExplicitly, the four pairs areSince $AB=-BA$ and $AB\ne 0$, the group is noncommutative. More precisely, $C$ has order four, $A$ has order two and $ACA=C^{-1}$, identifying $G$ with the dihedral group of a square. This fixes the convention unambiguously: it is the dihedral group of group order eight, irrespective of whether it is denoted $D_4$ or $D_8$.