TITLE: Example of nonabelian group with all proper subgroups normal
QUESTION [12 upvotes]: 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

REPLY [12 votes]: 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.