Group Theory

Group theory is a fascinating study of systems and groups. It studies transformations that lead to some sort of symmetry. The field might be a little abstract but bear with me as I try to explain it.

What is a Group?

A group required a set of elements (G) and an operation $(\cdot )$. For something to be considered a group, it must satisfy 4 conditions.

Closure

A group has to be closed. Thus $\mathrm{\forall }a,b\in G$.

Associative

A group has to satisfy associative properties. Thus $\mathrm{\forall }a,b,c\in G$.

Identity

A group has to satisfy Identity properties. Thus, there $\mathrm{\exists }e\in G$ where $\mathrm{\forall }a\in G$

Inverse

A group has to satisfy Inverse properties. Thus $\mathrm{\exists }b\in G$ where $\mathrm{\forall }a\in G$.

Understand Logic notation Notation

Examples of a Group

Why is it a Group?

• Closure
  • The addition of two integers logically results in another integer.
• Associative
  • The addition of integers is not dependent on the order in which it is applied.
• Identity
  • The identity element of $\mathbb{Z}$ is $0$.
• Inverse
  • The inverse $\mathrm{\forall }a\in \mathbb{Z}$ is $-a$.

Why is it not a Group?

• Closure
  • The multiplication of any integers always results in another integer
• Associative
  • It is independent of the order in which it is applied
• Identity
  • The identity of this system is 1.
• Inverse
  • It does not have an inverse. since inverse of $a$ is $\frac{1}{a}$. But $\mathbb{Z}$ is for integers only. Not fractions.

Commutativity

A group which satisfy commutative properties are considered abelian. such that $\mathrm{\forall }a,b\in G$

Group theory finds transformations on elements in a set that preserves a certain structure and symmetry within the group. Read more here: Structures (Still working on it)

Isomorphic