Groups of order 1

This article gives information about, and links to more details on, groups of order 1
See pages on algebraic structures of order 1 | See pages on groups of a particular order

There is, up to isomorphism, a unique group of order 1, namely the trivial group. In fact, it is, up to isomorphism, the unique magma of order 1. We can see this by noting that the multiplication table on any one-element set is completely constrained.

Statistics at a glance

Quantity	Value	Explanation
Total number of groups	1	Only one possible multiplication table
Number of abelian groups	1	Identity times identity is identity
Number of simple groups	0	The trivial group is not considered simple by convention - a simple group needs precisely 2 normal subgroups, analogous to how 1 is not a prime number
Number of nilpotent groups	1	Trivial derived series. Only group of nilpotency class 0.