Problem 3

State the number of elements in $\mathrm{S}_{4}$.

1. Let $G$ be the cyclic group of order 4. Cayley’s theorem (1.17) says that it is isomorphic to a subgroup of $\mathrm{S}_{4}$. Find one such subgroup of $\mathrm{S}_{4}$ and prove that it is, indeed, isomorphic to $G$.

2. Let $G$ be the group with presentation

   $$\langle a,b\mid a^{2}=b^{2}=(ab)^{2}=e\rangle.$$

   Cayley’s theorem says that it is isomorphic to a subgroup of $\mathrm{S}_{4}$. Find one such subgroup of $\mathrm{S}_{4}$ and prove that it is, indeed, isomorphic to $G$.