Show that every subgroup of the quaternion group Q is a normal subgroup of Q [closed]
How should I start to show this. Can anyone give the specific steps?
I'm assuming quaternion is $Q_8 = \{ \pm 1, \pm i, \pm j, \pm k \}$.
The trivial subgroups satisfies the normal condition trivially.
Note first that $\langle i \rangle= \{\pm 1, \pm i\}$ has order $4$, so the index $|Q_8: \langle i \rangle|= 2$. It follows that it is a normal subgroup. Same argument goes for $\langle j \rangle$ and $\langle k \rangle$
Last subgroup to check is $\{\pm 1\}$, but this is the center of the group, so is also a normal subgroup.