Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 10 / c / Solution

The three order-four subgroups areIn each case the generator has powers $A,A^2=\hat{W},A^3=\hat{A},A^4=W$, so all three are cyclic and mutually isomorphic to $C_4$, a cyclic group of order four. For $G_1$ the table isThe maps $X^j\mapsto Y^j$ and $X^j\mapsto Z^j$ are explicit group isomorphisms. These are all order-four subgroups: any such subgroup contains an element other than $W,\hat{W}$, and that element has order four and already generates the subgroup. They share the order-two subgroup $\{W,\hat{W}\}$ and are each normal, being of index two.