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.
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.
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.
Groups in which every subgroup is normal are sometimes called "Dedekind groups", with nonabelian Dedekind groups being called "Hamiltonian groups".
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 $N = \langle (1 2 3) \rangle \subset S_3$.