I'm preparing for an exam, and this little guy just had me stumped:
Let $G$ be a group, $N\subset G$ a normal subgroup of $G$, $x,y,z \in G$, and $x^3 \in N$, $y^5 \in N$, $zxz^{-1}y^{-1} \in N$. Show that $x,y\in N$.
What I want to do is to show for that for $g \in G$ that $gxg^{-1} \in N$. Another approach I tried is to just multiply the 3 given elements in $N$ in certain ways and use that $N$ is closed under multiplication to show that $x \in N$. No luck so far. This can't be that difficult right?