# Direct Sum and Isomorphism

• September 28th 2013, 10:08 PM
jcir2826
Direct Sum and Isomorphism
If, A,B, C are finite abelian groups then prove

(i) A direct sum B isomorphic to A direct sum C implies that B is isomorphic to C
(ii) A direct sum A isomorphic to B direct sum B implies A isomorphic to B.
• September 29th 2013, 05:06 AM
SlipEternal
Re: Direct Sum and Isomorphism
Use commutative diagrams:

$\begin{matrix}A\bigoplus B & \to & A\bigoplus C \\ \downarrow & \searrow & \downarrow \\ B & \to & C\end{matrix}$

You may need to make the diagrams bigger (include groups like $(A\oplus B) / (A \oplus \{e_B\})$, depending on how much detail you need).