Group
Definition of a Group:

Set and Operation: A group $(G,\cdot )$ consists of a set $G$ and a binary operation $\cdot $such that:
 Closure Property: For any $a,b$ in $G$, $a\cdot b$ is also in $G$.
 Associative Property: For all $a,b,c$ in $G$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
 Identity Element: There exists an element $e$ in $G$ such that for any $a$ in $G$, $a\cdot e=e\cdot a=a$
 Inverse Element: For each $a$ in $G$, there exists an element ${a}^{1}$ in $G$ such that $a\cdot {a}^{1}={a}^{1}\cdot a=e$.
Key Concepts:

Identity Element: Denoted as $e$ or $1$, it is the element that leaves other elements unchanged when combined. For any element $a$ in $G$, $a\cdot e=e\cdot a=a$.

Inverse Element: For each $a$ in $G$, there exists an element ${a}^{1}$ in $G$ such that $a\cdot {a}^{1}={a}^{1}\cdot a=e$.

Closure Property: For any $a,b$ in $G$, $a\cdot b$ is also in $G$. This means the result of the operation stays within the group.

Associative Property: The grouping of elements does not affect the result of the operation. For all $a,b,c$ in $G$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
Examples:

Integers with Addition: The set of integers $\mathbb{Z}$ with the operation of addition forms a group. The identity element is $0$ and the inverse of any integer $n$ is $n$.

NonZero Rational Numbers with Multiplication: The set of nonzero rational numbers $\mathbb{Q}\{0\}$ with the operation of multiplication forms a group. The identity element is $1$ and the inverse of any nonzero rational number $q$ is $\frac{1}{q}$.

Symmetric Group: The set of all permutations on a finite set forms a group called the symmetric group. The operation is the composition of permutations. The identity element is the identity permutation, and each permutation has an inverse, another permutation that undoes its operation.
Subgroups:

Definition: A subgroup $H$ of a group $G$ is a subset of $G$ that forms a group under the same operation as $G$.

Criteria: For $H$ to be a subgroup of $G$:
 $H$ must contain the identity element of $G$.
 $H$ must be closed under the operation in $G$.
 $H$ must contain the inverses of its elements in $G$.
Group Properties:

Unique Identity and Inverse: In a group, the identity element and the inverse of each element are unique.

Cancellation Law: In a group, if $a\cdot b=a\cdot c$, then $b=c$ and if $b\cdot a=c\cdot a$, then $b=c$.

Order of a Group: The number of elements in a group is its order. It can be finite or infinite.

Cyclic Groups: Groups generated by a single element are cyclic groups. If there exists an element $g$ in $G$ such that repeatedly applying the group operation generates all elements of $G$, then $G$ is cyclic.