Prove that a quotient group has an element of order 2. 
Let $H$, $K$ be subgroups of $G$, with $H$ being a normal subgroup of $G$. Suppose the cardinality of $H\cap K$ is odd and $K$ has an element $k$ of order $2$. How would you prove that the quotient group $G/H$ has an element of order $2$? 

I really don't understand this material well, would it be possible to be explained at an elementary level?

Here's a rare case where we can use the full power of the second isomorphism theorem. 
We prove that $HK/H$ has an element of order $2$. By the second isomorphism theorem, this is isomorphic to $K/(H\cap K)$. Since $K$ has an element of order $2$, $|K|$ is even by Lagrange's theorem. By assumption $|H\cap K|$ is odd, so $|K/(H\cap K)|=|K|/|H\cap K|$ is even. By Cauchy's theorem there is an element of order $2$, hence there is an element of order $2$ in $HK/H\subseteq G/H$.
