Paper 3, Section II, D

State Lagrange's theorem. Let $p$ be a prime number. Prove that every group of order $p$ is cyclic. Prove that every abelian group of order $p^{2}$ is isomorphic to either $C_{p} \times C_{p}$ or $C_{p^{2} \text {. }}$

Show that $D_{12}$, the dihedral group of order 12 , is not isomorphic to the alternating $\operatorname{group} A_{4}$.

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