Why $a^p\in N$ if $[a]^p=[e]$ holds in $G/N$

Suppose there exists an element in $G/N$ of order $p$, where $p$ is a prime number. In other words, there exists an $a\in G$ such that $[a]^p=[e]$, where $[a]\neq[e]$.

Then why $a^p$ belongs to $N$?

I understand if $a^p$ belongs to $N$, then since $[a^p]=Na^p$, by closure of $N$, $[a^p]$ is a subset of $N$; Since $N=Na^{-p}a^p$, by the fact that $N$ is closed under inversion and multiplication, $N$ is a subset of $[a^p]$, thus $[a^p]=N=[e]$. But although the statement that $a^p$ belongs to $N$ leads to the fact that $[a]^p=[e]$, it's not a necessary condition for these two to be equal.

• Please, try to make the title of your question more informative. E.g., Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. Apr 2 '14 at 14:14
• oops, sorry, i'll try my best Apr 3 '14 at 0:59

By definition, $[a]^p=[a^p]$ and saying $[a]^p=[e]$ becomes $$[a^p]=[e]$$ or $a^p\in N$, because the equivalence relation you're using is $$x \sim y \text{ if and only if } xy^{-1}\in N.$$ In different notation, where we use $[x]=Nx$, $$Nx=Ny\text{ if and only if }xy^{-1}\in N.$$
$[a^p] = [a]^p = [e] \iff a^pN = eN = N$. We also have $a^p = a^p\cdot1 \in a^pN = N$, so $a^p \in N$.