11
$\begingroup$

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

  • 0
    Can we give an example other than the quaternions?2014-04-06

1 Answers 1

11

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.

  • 1
    @user42912 Because $aZa^{-1}$ is a subgroup of order $2$ for every $a \in Q$.2014-03-31