[Image 1]
Hey it's a me again @drifter1!
Today we continue with Mathematics, and more specifically the branch of "Discrete Mathematics", in order to get into Groups.
So, without further ado, let's get straight into it!
Groups are algebraic structures studied by the branch of mathematics known as Group Theory. A group (G, *) is defined as a non-empty set G of elements together with a binary operation (also called group operation) on that set, which satisfies the following 4 properties:
The first property is commonly implied by defining the binary operation as a function, and so the number of properties is often 3 instead of 4.
The properties or features of groups yield additional theorems.
First of all, the identity element is unique. It's easy to prove this. Choosing e and e' to be two identities in G results in ae = a and ae' = a. And so e = e', and there is only one identity element.
In a similar manner, there is also only one inverse for each element, meaning that the inverses are unique. Additionally, the inverse of the inverse equals the original element.
Due to associativity and identity, the left and right cancellation laws are also satisfied. This allows for simplifications, where the same quantity is removed on both the left-hand side and right-hand side.
Groups can be finite and infinite. A group (G, *) is a finite group when G is a finite set. Similarly, a group is an infinite group when G is a infinite set. The set of integers together with the addition operation form a valid infinite group.
For a finite group, the group order is the number of elements it consists of. This is commonly denoted by |G|.
For example, a group of order 1 consists only of the identity element e. This special group is known as a trivial group. Similarly, a group of order 3 consists of the identity element and two other elements.
Groups which also satisfy the commutativity property (a * b = b * a) are known as Abelian Groups.
Another common group type is the so called Cyclic Group. In such a group each element can be written in the form xn for some integer n and generator x.
Groups can be multiplied in order to create more groups. This is commonly done through a direct product. For groups G and H it can be defined as:
The corresponding group operation is:
A group homomorphism is a mapping function f : G → H, which preserves the group operation, even when the binary operations of G and H are different. In other words, the identity property and group operation are still satisfied.
For a homomorphism there might also exist the inverse mapping, known as a isomorphism. In such a case, the two groups are isomorphic and so basically the same.
Mapping from one group to another group where the set is the same but the binary operation is different, results in a mapping known as an automorphism.
Mathematical equations used in this article, have been generated using quicklatex.
Block diagrams and other visualizations were made using draw.io.
And this is actually it for today's post!
Next time we will continue on with more on Group Theory...
See ya!
Keep on drifting!
