Archives

# 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!

Related Posts

### Comparing Topologies

The Back Pocket

### Need to Prove Your Ring is NOT a UFD?

The Back Pocket

### Operator Norm, Intuitively

The Back Pocket

### Four Flavors of Continuity

The Back Pocket

Leave a comment!