Group Theory

A Group is a Non-Empty Set with following Additional Properties
1. Binary Operation/Operator Property: Every Group supports a Single Binary Operation that can be Performed Between any 2 Elements of the Group The actual Binary Operation/Operator supported depends upon the Group itself and may vary from one Group to other.
2. Closure Property: Every Group is Closed under its Supported Binary Operation. This means that Performing the Supported Binary Operation between any 2 Elements of a Group gives us an Element that is also a part of the Group.
  
  Any Non-Empty Set that supports a Binary Operation between any 2 of its Elements and has the Closure Porperty as mentioned above is called a Magma or a Groupoid.
3. Associative Property: The Binary Operation supported by any Group is Associative. For example, if \(e_1\), \(e_2\) and , \(e_3\) are Elements of any Group which supports a Binary Operation denoted by \(\circ\) then
  
  \(e_1 \circ (e_2 \circ e_3) = (e_1 \circ e_2) \circ e_3\)
  
  Any Magma / Groupoid that supports the Associative Porperty as mentioned above is called a Semi-Group.
4. Identity Property: Every Group has a Unique Identity Element. Performing the Binary Operation between the Identity Element and Any Other Element in the Group gives back the Element itself. For example, if \(e_1\), \(e_2\) and , \(e_3\) are Elements of any Group which has the Identity Element \(e\) and supports a Binary Operation denoted by \(\circ\) then
  
  \(e_1 \circ e = e \circ e_1 =e_1\)
  
  \(e_2 \circ e = e \circ e_2 =e_2\)
  
  \(e_3 \circ e = e \circ e_3 =e_3\)
  
  Any Semi-Group that supports the Identity Porperty as mentioned above is called a Monoid.
5. Inverse Property: Each Element in every Group has a Corresponding Unique Inverse Element. Performing the Binary Operation between Any Element and its Inverse Element gives the Identity Element. For example, if \(e_1\), \(e_2\) and , \(e_3\) are an Elements of any Group which has the Identity Element \(e\) and supports a Binary Operation denoted by \(\circ\) then the Group will also contain the Elements \({e_1}^{-1}\), \({e_2}^{-1}\) and , \({e_3}^{-1}\) (Inverses of Elements \(e_1\), \(e_2\) and , \(e_3\) respectively) such that
  
  \(e_1 \circ {e_1}^{-1} = {e_1}^{-1} \circ e_1 =e\)
  
  \(e_2 \circ {e_2}^{-1} = {e_2}^{-1} \circ e_2 =e\)
  
  \(e_3 \circ {e_3}^{-1} = {e_3}^{-1} \circ e_3 =e\)
  
  Any Monoid that supports the Inverse Porperty as mentioned above is called a Group.
  
  As a corollary to the Inverse Property, for any Group having Elements \(e_1, e_2, e_3, ... , e_n\) and having Binary Operator \(\circ\)
  
  \({(e_1 \circ e_2 \circ e_3 \circ \cdots \circ e_n)}^{-1}= {e_n}^{-1} \circ \cdots \circ {e_3}^{-1} \circ {e_2}^{-1} \circ {e_1}^{-1}\)
Please Note that Although the Binary Operation between any 2 Elements of a Group May or May Not be Commutative, the Binary Operation is always Commutative between the Identity Element and Any Other Element of the Group as well as Any Element and its Inverse.
A Group is represented by a Combination of a Set (denoted by some Alphabet/Alphanumeric Symbol) and the Binary Operation that it supports (denoted by some Symbol that Represents a Binary Operator).
Any Group in which Binary Operation between all pairs of any 2 Elements of the Group are Commutative is called an Abelian Group. Any Group in which Binary Operation between atleast one pair of any 2 Elements of the Group is Non-Commutative is called an Non-Abelian Group.
A Group containing Countable (i.e. Finite Number) of Elements is called a Finite Group. A Group containing Uncountable (i.e. Infinite Number) of Elements is called an Infinite Group.
The Number of Elements present in a Finite Group is called the Order of the Group. Given any Group \(A\), the Order of the Group is denoted by \(|A|\).

Please Note that All Groups having Order \(\leq 5\) are Abelian. For any Group to be Non-Abelian it must have an Order \(\geq 6\).
Binary Operation on an Element of a Group with itself is represented as the Element raised to the Power of a Positive Integer Exponent. The value of the Positive Integer Exponent / Power is One more than the Number of times the Binary Operation has been performed. For example, for any Element \(e_1\) in any Group having Binary Operator \(\circ\), following gives the representation of Binary Operation on an Element of a Group with itself

\(e_1 \circ e_1={e_1}^2\)    (Binary Operation Performed Once)

\(e_1 \circ e_1 \circ e_1={e_1}^3\)    (Binary Operation Performed Twice)

\(e_1 \circ e_1 \circ e_1 \circ e_1={e_1}^4\)    (Binary Operation Performed Thrice)
The Order of any Element in a Group is One more than the Minimum Number of times Binary Operation must be performed on the Element with itself (which is same as the Value of the Minimum Power / Positive Integer Exponent that any Element must be raised to) to get the Identity Element. For example, for any Element \(e_1\) in any Group having Binary Operator \(\circ\) and Identity Element \(e\), the Order of the Element is the \(N\) if

\(e_1 \circ e_1 \circ e_1 \circ \cdots (Binary\hspace{.1cm}Operation\hspace{.1cm}Performed\hspace{.1cm}N-1\hspace{.1cm}times)= {e_1}^N=e\)

Please Note that Order of Elements in Finite Groups is always Finite whereas Order of Elements in Infinite Groups Can be Finite or Infinite (i.e. Not Defined).
Any Group \(H\) is said to be a Subgroup of a Group \(G\) if
1. They support the same Binary Operation, have the same Identity Element and All Elements of Group \(H\) are also Present in Group \(G\).
2. For any Element \(e_1\) Present in the Group \(H\), the Element \({e_1}^{-1}\) (i.e Inverse of Element \(e_1\)) is also Present in \(H\).
3. For any 2 Elements \(e_1\) and \(e_2\) Present in the Group \(H\), the Elements \(e_1 \circ e_2\) and \(e_2 \circ e_1\) are also Present in \(H\) (where \(\circ\) is the Binary Operator Supported by the Group).
The following example demonstrates how Group-Subgroup Relations are represented

\(H \lt G\)   (Represents Group \(H\) is a Proper Subgroup of Group \(G\))

\(H \not \lt G\)   (Represents Group \(H\) is Not a Proper Subgroup of Group \(G\))

\(H \leqslant G\)   (Represents Group \(H\) is Either a Proper Subgroup of or Equal to Group \(G\). Also said as Group \(H\) is a Subgroup of Group \(G\)))

\(H \not\leqslant G\)   (Represents Group \(H\) is Niether a Proper Subgroup of Nor Equal to Group \(B\). Also said as Group \(H\) is Not a Subgroup of Group \(G\))

Given any Finite Group \(G\) and any of it's Subgoup \(H\), the Order of \(H\) is always a Divisor of Order of \(G\). This is known as Lagrange Theorem of Groups.

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