This has eluded me, and my professor didn't really explain it well. I understand that given that N is a normal subgroup, that G/N = gN such that g is in G. However I don't really see how this can make N the identity element of G/N
How does a quotient group (G/N for example) make the normal subgroup (N) the identity element?
0
$\begingroup$
abstract-algebra
group-theory
1 Answers
1
The identity element in $G/N$ is the class $eN$, which is simply $N$.
More generally, you can show that $$hN=eN \Leftrightarrow h \in N$$
This means that all elements in $N$, and only elements of $N$, become the identity in $G/N$.
-
0Oh, so it's because when multiplying any element in N by the entire subgroup, N is the result, so therefore any nN is always just going to be N, and therefore any element of n is always going to return N when you multiply it together by the entire subgroup? – 2017-02-28
-
0@user3491700 Exactly. Moreover, if $hN=N$, then since $e \in N$ you get $he \in hN=N$. This also gives you the converse of your statement. – 2017-02-28