It is my personal belief that group theory must be taught in high school mathematics. What do I mean by this? I believe that the field of group theory is so elementary to diverse factions of mathematics such as abstract algebra or number theory that it is necessary for secondary education to cover the concept.
Hence, considering the importance of the concept of the definitions of groups, rings, and fields, I felt that it would be necessary to set the foundation for more complex concepts that will be covered in my blog.
But before we delve into the intriguing world of groups, it would be impossible to do so without discussing what binary operators are.
A binary operator is basically any operator that gives us a value when two values are inputted. So addition or multiplication, or even logic gates would be examples of such, while more complicated examples such as
f(x,y)=x^2+xy-2y^2+x-y+3
(which is an interesting hyperbolic function by itself) would also be a binary operator.
So, what is a group?
A group, in essence, is a set with a binary operator that satisfies the four preconditions.
1. Closure: when the binary operator acting upon any two elements of the set produce an element of the set. In other words, for a group (X, *), the following holds true:
2. Associativity:
3. Identity:
Note: This identity element is unique.
4. Inverse:
An example of such would be the group of rational numbers-{0} under multiplication! If we multiply two rationals, the outcome is a rational number. Also, the identity element would be 1 and the inverse of p/q would be q/p. The cardinality of this group is infinity. Cardinality refers to the number of elements in the group.
There are also groups with finite cardinalities, such as the unitary group of 10, which is {1,3,7,9} with multiplication under modulo 8. The proof can be replaced with a Cayley table as below, which is a table representing the product of all possible combinations.
An extremely, extremely important clarification to make here is that groups need not only be comprised of numbers. Groups such as Dihedral groups are solely constructed of geometrical shapes.
If a*b=b*a as the unitary group of 10, we also call such groups Abelian groups. At first, the notion of groups may seem abstract and unimportant; yet, the more you read and study about the field, I bet that you will be amazed by the widespread use of groups.
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