Nilpotent Group - Examples

Examples

• As noted above, every abelian group is nilpotent.
• For a small non-abelian example, consider the quaternion group Q₈, which is a smallest non-abelian p-group. It has center {1, −1} of order 2, and its upper central series is {1}, {1, −1}, Q₈; so it is nilpotent of class 2.
• All finite p-groups are in fact nilpotent (proof). The maximal class of a group of order pn is n - 1. The 2-groups of maximal class are the generalised quaternion groups, the dihedral groups, and the semidihedral groups.
• The direct product of two nilpotent groups is nilpotent.
• Conversely, every finite nilpotent group is the direct product of p-groups.
• The Heisenberg group is an example of non-abelian, infinite nilpotent group.
• The multiplicative group of upper unitriangular n x n matrices over any field F is a nilpotent group of nilpotent length n - 1 .
• The multiplicative group of invertible upper triangular n x n matrices over a field F is not in general nilpotent, but is solvable.