# Help on Subgroup and Normal Subgroup

We know that a subgroup N of an abelian group G must be normal. However, is the reverse necessarily true? An illustration would enlighten me.

Any resource links or names of illustrative texts welcome, as I am new to Group Theory.

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It is not even true that if every subgroup of $G$ is normal then $G$ must be abelian. The smallest example is the quaternion group of order $8$, $Q_8 = \{\pm 1,\pm i,\pm j,\pm k\}$. The subgroup of order $4$ is normal because it has index $2$; the only subgroup of order $2$ is $\{\pm 1\}$, which is normal because it equals the center of the group.
I would like to add to this, since I was intrigued by this problem today. The structure of Hamiltonian groups was described by Baer as follows: A group $G$ is Hamiltonian iff $G\cong C\oplus T\oplus Q_8$ where $C$ is a direct sum of cyclic groups of order 2, $T$ is a torsion subgroup with only odd torsion. So every Hamiltonian group contains $Q_8$ in a very fundamental way, in that the quotient $G/Q_8$ is abelian (where $Q_8$ is embedded in the natural way) – you Oct 19 '12 at 11:57
By the converse, I assume you mean if a subgroup $N$ of a group $G$ is normal, must $G$ be abelian? If so, this is definitely not true, and is a reason why normal subgroups are interesting and important. The simplest example of a normal subgroup in a non-abelian group is the subgroup $N = \langle (1 2 3) \rangle \subset S_3$.