Some group theory related concepts would go here, but only the things that I need right now (JIT learning):
Group
A group is a set G with a binary operation denoted by a dot (. or ) that combines any two elements in the set to produce a new third element within the same set and the following conditions must hold:
- the operation is associative, i.e. where is any binary operation.
- there exists an element in G such that for every in G, one has . Such an element, is unique and is called the identity element.
- for each in G there exists an element in G such that . Such an element is unique and is called the inverse of , commonly denoted as .
Abelian Group
A group in which the result of applying the group operation (say ) to two group elements does not depend on the order in which they are written. That is, the group is commutative in nature.
Thus, an abelian group is a group in which .
Note that the condition for an abelian group is different from the third condition for a group, as in the third condition we are saying that an inverse exists for each in G, while here we are saying that we can pick any two random pairs in G and still have , where is not necessarily the inverse of .
ex. is an abelian group because we can pick any two elements in the group of integers and it follows that a + b = b + a, and also holds other group conditions, ex. every integer n has an additive inverse -n. The identity element in addition is 0.
The ability of taking two elements from a set and getting a third element after a binary operation that’s still within the same set is called closure. If you look closely, it too is one of the conditions for a group. In closure holds as when we take a pair of integers and add them we always get another integer.