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
Four Flavors of Continuity
The Back Pocket
Operator Norm, Intuitively
The Back Pocket
English is Not Commutative
The Back Pocket
Motivation for the Tensor Product
The Back Pocket
Leave a comment!