Helpful topic to teach yourself lattice-based crypto: groups

In another article about rings and fields, I briefly touched on the topic of an abelian group.

As a quick refresher, in order for a set that happens to be equipped with a binary operation to even be considered an abelian group, it must satisfy the following properties:

• Closure: $a + b$ is a part of the field. (For example, adding two numbers together will always get another number.)
• Associativity: $(a + b) + c = a + (b + c)$
• Identity: there exists an element $O$ such that $a + O = a$
• Inverses: for every $a$, there exists some $d$, such that $a + d = O$
• Commutativity: $a + b = b + a$

But a generalization of an abelian group is just a group, and it merely involves removing the requirement that a set equipped with a binary operation to not guarantee to be commutative.

So in other words, get rid of the commutativity requirement, and we're left with a group.

Groups are an important topic to explore, because later on, we are going to be exploring the idea of a subgroup.