Paper 3, Section I, D

State Lagrange's Theorem.

Let $G$ be a finite group, and $H$ and $K$ two subgroups of $G$ such that

(i) the orders of $H$ and $K$ are coprime;

(ii) every element of $G$ may be written as a product $h k$, with $h \in H$ and $k \in K$;

(iii) both $H$ and $K$ are normal subgroups of $G$.

Prove that $G$ is isomorphic to $H \times K$.

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