Let $K,M,N$ be groups such that $K$ is finite and $K \times M$ is isomorphic to $K \times N$. Prove that $M$ and $N$ are isomorphic
My initial thoughts: I believe I should use induction on the order of $K$ because it is finite. I would then take $G=K_1H_1=K_2H_2$ with $K_1,H_1,K_2,H_2$ are all normal subgroups of $G$ such that $K_1 \cap H_1 = K_2 \cap H_2 = \{e\}$. Additionally, we take $K_1$ and $K_2$ are both isomorphic to $K$. Now take $M$ and $N$ such that $H_1$ is isomorphic to $M$ and $H_2$ is isomorphic to $N$. We then take $T_1=K_1 \cap H_2$ and $T_2=K_2 \cap H_1$. I believe it might be easier to show that $K_1/T_1 \times K_2/T_2 \times H_1$ and $K_1/T_1 \times K_2/T_2 \times H_2$ are isomorphic. My knowledge of quotient groups might be a little weak so I am stuck around here.
