understand quotient group i am trying to understand what does mean  quotient terminology in group theory  by as simple way as possible,also quotient group  i want to know something about it,using internet i read that
"
In mathematics, specifically group theory, a quotient group (or factor group) is a group obtained by aggregating similar elements of a larger group using an equivalence relation. For example, the cyclic group of addition modulo n can be obtained from the integers by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity."
so does it means that quotient  group helps us to divide element into equivalence classes?
example is there :
For example, consider the group with addition modulo 6:
G = {0, 1, 2, 3, 4, 5}.
Let
N = {0, 3}.
The quotient group is:
G/N = { aN : a ∈ G } = { a{0, 3} : a ∈ {0, 1, 2, 3, 4, 5} } =
{ 0{0, 3}, 1{0, 3}, 2{0, 3}, 3{0, 3}, 4{0, 3}, 5{0, 3} } =
{ {(0+0) mod 6, (0+3) mod 6}, {(1+0) mod 6, (1+3) mod 6},
{(2+0) mod 6, (2+3) mod 6}, {(3+0) mod 6, (3+3) mod 6},
{(4+0) mod 6, (4+3) mod 6}, {(5+0) mod 6, (5+3) mod 6} } =
{ {0, 3}, {1, 4}, {2, 5}, {3, 0}, {4, 1}, {5, 2} } =
{ {0, 3}, {1, 4}, {2, 5}, {0, 3}, {1, 4}, {2, 5} } =

{ {0, 3}, {1, 4}, {2, 5} }.

as i know notation $G/N$  to be the set of all left cosets of N in G, i.e., G/N = { aN : a in G }. 
so what does given  result represent  to original  groups?thanks in  advance,i was also interested about this topic because of  Homology,which as i understand is related to cycles and boundary in group
 A: It is important to keep in mind that to have in the quotient set $G/N$ the structure of a group, $N$ has to be normal. 
But in general one can construct the quotient set even in the case that $N$ wouldn't be normal, and in this general case the set $G/N$ is the set of cosets which allow you to partition the group into parts, the cosets, which all of them have the same cardinality, an equipartition. 
A: Your summary of $G/\{0, 3\}$ is exactly right; there are three elements of the quotient, and each is a two-element set. 
There remains the question of "how do you define addition on the quotient"? There are two strategies:


*

*Define $A + B = \{a + b : a \in A \text{ and }b \in B\}$. You need to then prove that if $A$ and $B$ are cosets, so is $A+B$. 

*Define $A + B$ by saying "Pick $a \in A$ and $b \in B$, and define $A+B$ to be the coset containing $a+b$." You then need to show that the result is independent of your choice of $a$ and $b$. 
Both approaches work fine. 
Finally, having defined addition, you need to show it's actually an operation with inverses, identity, associativity, but those are relatively easy. 
