#abelian groups

[isomorphismes]

In a short exact sequence of vector fields 0 → A → B → C→ 0,B must be the direct sum B=A⊕C of the subgroup A and the quotient group C=B/A.

Allen Hatcher

[isomorphismes]

the stable homotopy category by Cary Malkiewich

[studyingwithjessie]

Fundamental Theorem of Abelian Groups

[tunnels-end]

Can we talk about how Tarski managed to elegantly compress all 4 axioms of Abelian groups into just 2?

[eatsleepmath]

The colimit of subgroups of \(\mathbb{Z}\)

Compute the colimit (in the category of abelian groups) of the diagram \[\mathbb{Z} \overset{2!}{\to} \mathbb{Z} \overset{3!}{\to} \mathbb{Z} \dots\] where the map \(\mathbb{Z} \overset{n}{\to} \mathbb{Z}\) factors into \(\mathbb{Z} \to n \mathbb{Z} \hookrightarrow \mathbb{Z}\). The answer's \(\mathbb{Q}\), it takes a little work, in general you just need your collection of transition maps to intersect every nontrivial subgroup of \(\mathbb{Z}\), because then you can specify where each \(\frac{1}{n}\) in \(\mathbb{Q}\) gets sent to to define the unique mediating morphism.