Group theory and its examples

Group theory studies abstract structures called groups. Algebraic structures such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups have been applied to a wide range of other mathematical structures, some of the applications of groups are:
  1. Cryptography 
  2. Galois theory 
  3. Algebraic topology 
  4. Algebraic geometry 
  5. Algebraic number theory 
  6. Combinatorics 
  7. In chemistry it has been used to study crystals and physical atoms.
Rubik cube has applications in mathematical group theory, the construction and solutions of Rubik cube can be referenced to the theory of groups.

Mathematical Definition of Groups
A group can simply be defined as a set paired with a binary operation that satisfies the group axioms.  

A group is a system $(G,\circ)$ consisting of a non-empty set $G$ and a binary operation $\circ$ satisfying the following axioms:
  1.  Closure: If $a\in{G}$ and $b\in{G}$ then $a\circ{b}\in{G}$. 
  2. Associativity: If $a,b,c\in{G}$ then $a\circ(b\circ{c})=(a\circ{b})\circ{c}$
  3.  Existence of identity: If $a\in{G}$ then there exists an element $e\in{G}$ called an identity element such that $e\circ{a}=a\circ{a\in{G}}$
  4.  Existence of an inverse: If $a\in{G}$ there exists an inverse element $a^{-1}\in{G}$ such that $a\circ{a^{-1}}=e$ where $e\in{G}$ and $e$ is an identity element.
If the axiom of commutation is satisfied in addition to closure, associativity, identity and inverse axioms, then such group becomes an Abelian group or commutative group.
  1. Commutativity: If $a,b\in{G}$ then $a\circ{b}=b\circ{a}$, $e\circ{a}=a\circ{e}=a$ and $a\circ{a^{-1}}=a^{-1}\circ{a}=e$.
We can see from the closure axioms that, for any finite number of elements in a group, if the group operation is defined on the elements, then the result will also be an element in the group. 

Example 
  1.  If $\mathbb{Z}$ is a set of integers with operation $+$ then $\mathbb{Z}$ is a group. Let $2,3$ and $4$ be elements in $\mathbb{Z}$ then $2+3+4=9\in\mathbb{Z}$, this shows the existence of closure law. 
  2. The set $\mathbb{Z}^+$ with operation $"+" $ is not a group. Reason: There is no identity element for positive integers. 
  3.  The set of all non-negative integers that includes $0$ with operation $"+"$ is not a group. $0$ is an identity element, but there is no inverse for other numbers $n>0$ where $n\in\mathbb{Z}$. 
  4. The set $\mathbb{Z}^+$ with operation $"×"$ is not a group. There is an identity element $1$ but no invsrse for all the numbers, becuase the numbers are non-negative numbers. 
  5. The following sets of numbers with operation $"+"$ are all groups namely: $(\mathbb{R},+),(\mathbb{Q},+),(\mathbb{C},+)$ are all groups. 
Read more on groups: Wikipedia - Group theory
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