Binary operation (on a set G)
A function that assigns to each ordered pair of elements of G an element of G
Operation table, a.k.a. Cayley table
A table of results of a binary operation (written in the form of a multiplication table)
Closure
The condition that members of an ordered pair from a set G combine to yield a member of G
Group
A set G together with a binary operation for which the following properties are satisfied:(1) Associativity: (ab)c = a(bc) for all a, b, c in G,(2) Identity: there exists an element e in G for which ae = ea = a for all a in G,(3) Inverses: For each element a in G, there is an element b ∈ G for which ab = ba = e
Commutative, a.k.a. Abelian (pertaining to a group G)
Having the property that ab = ba for all a,b ∈ G
Uniqueness of the Identity
A group contains only one identity element
Cancellation
In a group, left and right cancellation hold,Left: ab = ac implies b = cRight: ba = ca implies b = c
Uniqueness of inverses
For each element a ∈ G, there is a unique b ∈ G for which ab = ba = e.