Quotient groups (Work in progress post)

since I'm doing some prep for algebra, let me explain a little about quotient group and the quotient map. i encountered this in MA1100, and MA2101, but tbh i never really fully understood it. until algebra. super fun, super cool. so hopefully, i can do a good enough job of explaining it simply, such that those who like me didn't get it in MA1100 and MA2101 would not be so lost.

ok let's use this diagram to explain.

So, we start with $f:G\rightarrow H$, as shown by the top arrow. $f$ maps items from $G$ to $H$.

$f$ is surjective, could be injective or not. 

$q_{a}:G\rightarrow G/A$, maps items from $G$ to its equivalence class in $G/A$

$f_{1}: G/A \rightarrow H$ maps the equivalence classes in $G/A$ to $H$. in this case, the equivalence classes in $G/A$ are the elements. equivalence classes aka cosets. 

example (which i invented hehe, suppose all tans stays in the west, lees in the east, ngs in the south)

$G = \left \{ Tan Mark, Tan John, Lee Matthew, Ng Luke \right \}$

$H = \left \{ stay in the West, stay in the East, stay in the South \right \}$

$q_{a} = \left \{ surname Tan, surname Lee, surname Ng \right \}$

so under this map, $f:G\rightarrow H$, $f(Tan Mark)=stay in the West$, $f(Tan John)=stay in the West$, $f(Lee Matthew)=stays in the East$, $f(Ng Luke)=stays in the south$

then $q_{a}:G\rightarrow G/A$ maps each ppl to their surnames, i.e $q_{a}(Tan Mark) = surname Tan$, $q_{a}(Ng Luke) = surname Ng$ etc . this group is definitely surjective, because each element has to belong to some class, they all have a surname, so definitely surjective. 

then $f_{1}: G/A \rightarrow H$ maps surnames to where they stay for eg. wah damn tiring to type all i v tired

but it's that simple, nothing tricky about it. as prof says, "the quotient map doesn't tell u much, plus, we're not even doing much to get it. it just falls out if you stay at it hard enough. "

well there's a little more group axioms in the quotient group itself too. and isomorphism. screw it, i lazy explain now *LOL* i'm so not a dedicated math blogger *LOL*