Definition: A subgroup $N$ is a proper normal subgroup of a group $G$ IFF $Ng=gN \; \; \forall g \in G$
A proper normal subgroup $N$ of $G$ is a normal subgroup $N$ of $G$ that is not the whole of $G$. Wouldn't this implies that there is at least one element $g$ in $G$ that is not in $N$? If so, doesn't this contradicts the definition of a normal subgroup?
Clarification is appreciated.
Thanks in advance.
Let $G$ be a group.
A subgroup $H$ is a subset of $G$ that is also a group (under the same operation). Notation: $H\le G$. For example, we always have $G\le G$ and we always have $1\le G$ (where $1$ denotes the trivial subgroup of $G$ consisting only of its neutral element).
A proper subgroup of $G$ is a subgroup $H$ of $G$ with $H\ne G$. Notation: $H<G$.
A normal subgroup is a subgroup $N\le G$ such that $\forall g\in G\colon gN=Ng$. Notation: $N\trianglelefteq G$. Note that we always have $G\trianglelefteq G$ and $1\trianglelefteq G$
A proper normal subgroup is a normal subgroup that is also a proper subgroup. Notation: $N\lhd G$. From the above examples, we see that at least for non-trivial groups $G$, there always axists at least on proper normal subgroup (namely the trivial subgroup).
(Actually, there is a similar possible cause of confusion as with e.g. $\subset$, where some authors use $\subseteq/\subset$ and others use $\subset/\subsetneq$)