Normality and the operation on cosets

#group_theory

Let $G$ be a group and $N$ a normal subgroup of $G$ . Let $a N$ denote the left coset defined by $a \in G$ , and consider the binary operation

G / N \times G / N \to G / N

given by $(a N, b N) \mapsto a b N$ .

Show the operation is well defined.
Show the operation is well defined only if the subgroup $N$ is normal.

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L A T E X

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Let $G$ be a group and $N$ a normal subgroup of $G$. Let $aN$ denote the left coset defined by $a\in G$, and consider the binary operation
\[
	G/N\times G/N\to G/N
\]
given by $(aN, bN)\mapsto abN$.
\begin{enumerate}[label=\alph*)]
	\item Show the operation is well defined.
	\item Show the operation is well defined only if the subgroup $N$ is normal.
\end{enumerate}