# 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!