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Jack Rod
Jack Rod
08:41
@robjohn sir hi
actually I was reading about quotient ubgroup
 
1 hour later…
robjohn
robjohn
10:00
@JackRod is there a question?
Jack Rod
Jack Rod
@robjohn sir what is quotient group
the basic theorem of it
robjohn
robjohn
do you know what an equivalence relation is?
Jack Rod
Jack Rod
no
robjohn
robjohn
you should look them up because they are key in a quotient group
Jack Rod
Jack Rod
oh k
yes I know what is a equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation is equal to is the canonical example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. == Notation == Various notations are used in the literature to denote that two elements a {\displaystyle a} and b...
@robjohn
robjohn
robjohn
10:07
Do you know about modding by an equivalence relation?
Jack Rod
Jack Rod
no let me check for once
modular arithmetic, we say that two numbers are equivalent if they have the same remainder when divided by some number n. So, if we are working mod 4, we say that 1≡5. That does not mean that 1=5, just that they are equivalent for our purposes.
@robjohn
robjohn
robjohn
That is for one equivalence relation: being equal mod some number $n$. This can be done with any equivalence relation.
Jack Rod
Jack Rod
ok
robjohn
robjohn
This gives quotient rings, but the idea can be extended to groups. If you have a normal subgroup, $N$, you can form an equivalence relation by saying that $a\equiv b$ when $aN=bN$.
Modding by this equivalence relation is called modding by the normal subgroup $N$.
There is one element of the quotient group for each coset $aN$ for $a$ in the group.
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