Why Quaternion Group Have All Normal subgroups

Question

I'm studying about quaternion Group. Here my questionn is " why all subgroups of quaternion group " are normal. Is there any special reason ? That quaternion group have all normal subgroups. ??

Answer 1

I don't think there is a deep reason that all subgroups of the quaternion group $Q$ are normal, but checking them all is fairly simple.

Subgroups of $Q$ have order $1, 2, 4, 8$. Those of order $1, 8$ are trivially normal. Subgroups of index 2 -- that is, in this case, of order 4 -- are always normal. This leaves only subgroups of order 2, and the only one is $\{\pm 1\}$, since all other elements have order 4. But $\{\pm1\}$ is the center of $Q$, and therefore normal.

Answer 2

One can actually classify all groups of which all subgroups are normal.

Theorem Let $G$ be a group. Then all subgroups of $G$ are normal if and only if $G$ satisfies one of the two following conditions.
(i) $G$ is abelian;
(ii) There exist groups $A$ and $B$ such that

• $G \cong A \times Q \times B$, where $Q$ denotes the quaternion group of order $8$
• $A$ is an abelian group with the property that every element has odd order
• $B$ is an abelian group with $x^2=1$ for all $x \in B$.

For a proof see Theorem 12.5.4 of M. Hall Jr. The Theory of Groups, New York, 1959. A group is called hamiltonian if it satisfies condition (ii) of the theorem.