One Unspoken Rule of Algebra

Here's an algebra tip! Whenever you're asked to prove $$A/B\cong C$$ where $A,B,C$ are groups, rings, fields, modules, etc., mostly likely the The First Isomorphism Theorem involved! See if you can define a homomorphism $\varphi$ from $A$ to $C$ such that $\ker\varphi=B$. If the map is onto, then by the First Isomorphism Theorem, you can conclude $A/\ker\varphi=A/B\cong C$. (And even if the map is not onto, you can still conclude $A/B\cong \varphi(A)$.)  Voila!

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