Quotient group if and only if

Question

My textbook says

$uN=vN$ if and only if $v^{-1}$ such that $u \in \mathbb{N}$

What is the if and only if relation for right cosets $Nu=Nv$? If and only if $v \times u^{-1} \in \mathbb{N}$?

Answer 1

The following statements are equivalent:

• $Nu=Nv$
• $Nuv^{-1}=N$
• $uv^{-1}\in N$
• $N=Nvu^{-1}$
• $vu^{-1}\in N$