## Free Abelian Groups

Essentially, free abelian groups give us a rigorous way of talking about formal linear combinartions of some set of generators.

A collection of 6 posts

Essentially, free abelian groups give us a rigorous way of talking about formal linear combinartions of some set of generators.

It would be nice if there was some sort of relationship between cosets of the kernel and the image of a homomorphism. Oh wait... there is!

Let's now revisit the quotient set $G/H$, where $H$ is a subgroup of $G$. What we'd really like to do is turn $G/H$ into a group in a meaningful way. What should the group operation be, though?

It's a bit difficult to explain exactly why cosets are so important without working with them for a while first. But as you'll hopefully start to understand within my next few posts, cosets pop up everywhere and are a necessary tool to get anything done in the world of algebra.

A recurring theme in mathematics is that examining the maps between objects is indispensable to understanding those objects themselves. Of course, that depends on choosing the "correct" type of maps.

Essentially, a group is a set endowed with a very basic structure. This structure is enforced by an operation which governs how the elements in the group interact with each other.

- © 2019 Eric Shapiro