Nilpotent not implies abelian

Statement
A nilpotent group need not always be abelian.

Proof
The smallest examples of nilpotent non-abelian groups are of order 8 (see groups of order 8):


 * particular example::quaternion group, an eight-element group whose elements are $$\pm 1, \pm i, \pm j, \pm k$$ with the multiplication given by multiplication of quaternions.
 * dihedral group of order eight, generated by an element of order four and an element of order two that conjugates the element of order four to its inverse.

Converse
The converse is true: abelian implies nilpotent.

Reduction to prime power case for finite groups
Any group of prime power order is nilpotent, so any such group that is non-abelian furnishes an example of a nilpotent non-abelian group.

In a sense, the examples of prime power order are representative of all examples for finite groups, because of the following: By the equivalence of definitions of finite nilpotent group, a nilpotent group is the direct product of its Sylow subgroups. Thus, for a nilpotent group, if the group is an A-group (all Sylow subgroups are abelian) then it must be an abelian group. Thus, any example of a nilpotent non-abelian group ultimately relies on an example of prime power order.

Partial truth

 * Nilpotent of cube-free order implies abelian

Numerical comparison of number of abelian and nilpotent groups for finite orders
For any finite number $$n$$, we have Number of nilpotent groups equals product of number of groups of order each maximal prime power divisor. Similarly, the number of abelian groups is the product over all maximal prime power divisors of the number of unordered integer partitions of the exponent of each.

Thus, for any natural number $$n$$ such that there is at least one prime number $$p$$ for which $$p^3$$ divides $$n$$, there are nilpotent non-abelian groups of order $$n$$. For instance, for $$n = 144 = 2^4 \cdot 3^2$$, there are $$5 \times 2 = 10$$ abelian groups of that order, and $$14 \times 2 = 28$$ nilpotent groups of that order. See groups of order 144.