Originally Posted by

**deniselim17** I have 2 questions on cosets.

Let $\displaystyle G$ be a group with subgroup $\displaystyle \langle h \rangle$.

Let $\displaystyle g \in G \backslash \langle h \rangle$.

If there exists $\displaystyle N \lhd G$ such that $\displaystyle [a,h] \notin N$, show that $\displaystyle g \notin \langle h \rangle N$.

I have tried this problem for a long time, but I still can't get a convinced proof.

Please give me some hint.

I assume your "a" must in fact be "g", so $\displaystyle [g,h]\notin N\Longrightarrow g^{-1}h^{-1}gh\notin N$ , so if $\displaystyle g\in \langle h\rangle N$ then

$\displaystyle g=h^rn\,,\,r\in\mathbb{Z}\,,\,n\in N\Longrightarrow [g,h]=n^{-1}h^{-r}h^{-1}h^rnh=n^{-1}h^{-1}nh=[n,h]$ , but

this last rightmost expression is in N (why? Hint: normality), so we get a contradiction.

Another question is if $\displaystyle aN=bN$, is it true that $\displaystyle a=b$?

Not at all: $\displaystyle aN=bN\Longleftrightarrow b^{-1}a\in N$ and that's all.

For example, take $\displaystyle N=5\mathbb{Z}\,,\,a=2\,,\,b=7\Longrightarrow 2+N=7+N$

Tonio