M3. Elementary Group Theory:

Set is one of the most important concept in modern mathematics. It becomes more meaningful if we do something with or operate in its members or elements.

A group is a set $G$ together with an operation $*\; or \;•$ that combines any two elements $a$ and $b$ to form another element, denoted by $a*b$ or $a•b$ or $ab$.

» Algebraic Structure:

An algebraic structure consisting of a set $G$ under a operation $*$ on $G$, denoted by $(G,\; *)$. Let $a,b,c,d.............$ be the element of the set $G$, then algebraic structure (G, *) satisfies the following characteristics:

1) Closed:   If $a*b ∊ G$ for each $a, b ∊ G$.

2) Commutative:   If $a*b = b*a ∊ G$ for each $a, b ∊ G$.

3) Associative:   If $(a*b)*c = a*(b*c) ∊ G$ for each $a, b, c ∊ G$.

4) Existence of identity: For each $a ∊ G$, if there exists an element $e ∊ G$. Such that,

$a * e = e*a$; Where $e$ is called the identity element.
For additive, identity element $(e) = 0$ and for multiplication, identity element $(e) = 1$

For Example: If $x + 0 = x = 0 + x$ for all $x ∊ G$. Therefore $0$ is the identity element of $G$ for the operation $'+'$.
If $x*1 = x = 1*x$ for all $x ∊ G$. Therefore $1$ is said to be identity element of $G$ for the operation $'*'$.

5) Existence of inverse: For each $a ∊ G$, if there exists an element $a^{'} ∊ G$ such that,

$a*a^{'} = e = a^{'}*a$; Where $a^{'}$ is called the inverse element of $a$.

For Example: If $x +(-x) = 0$ for all $x ∊ G$. Then $-x$ is said to be inverse element of $G$ for the operation $'+'$.
If $x*x^{-1} = 1 = x^{-1}*x$ for all $x ∊ G$ and $x ≠ 0$. Then $x^{-1}$is said to be inverse element of $G$ for the operation $'*'$.

» Group (Group Axioms):

A group is a set $(G)$ with an operation$*$ denoted as $(G, *)$, Where $G$ is a non-empty set with an operation $'*'$. It is said to be group , if the operation $*$ satisfies the following Axioms:

G1) Closure Axiom: G is closed under the operation $*$,

i.e. $a * b ∊ G$ for all $a, b ∊ G$ .............................. (i)

G2) Associative Axiom: The binary operation $*$ is associative.

i.e. $(a * b)*c = a*(b*c)$ for each ∀ $a, b, c ∊ G$ ............... (ii)

G3) Identity Axiom: There exists an element  $e ∊ G$, such that

$a*e = a = e*a$, for all $a ∊ G$ .............................. (iii)

The element $e$ is called the identity of $'a'$ with respect to $'*'$ in $G$.

G4) Inverse Axiom: Each element of $G$ possesses inverse, (i.e. for each element $a ∊ G$, there exists an element $a^{'} ∊ G$), such that

$a * a^{'} = e = a^{'}*a$ .............................. (iv)

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