Example of nonabelian group with all proper subgroups normal
How do we find an example of nonabelian group for which all proper subgroups are normal?? It's one of the questions on my study-guide sheet. Thank you
Let $Q = \{1, -1, i, -i, j, -j, k, -k\}$ be the quaternion group. Let $Z$ be a subgroup of order $2$. Since $-1$ is the only element of order $2$, $Z = \{1, -1\}$. Since it is the only subgroup of order $2$, it is normal. Let $H$ be a subgroup of order $4$. Since $(Q : H) = 2, Q = H \cup aH = H \cup Ha$ for every $a \in Q - H$. Hence $aH = Ha$. Hence $H$ is normal.