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.

This article gives basic information comparing and contrasting groups of order 1. See also more detailed information on specific subtopics through the links:

Information type	Page summarizing information for groups of order 1
element structure (element orders, conjugacy classes, etc.)	element structure of groups of order 1
subgroup structure	subgroup structure of groups of order 1
linear representation theory	linear representation theory of groups of order 1 projective representation theory of groups of order 1 modular representation theory of groups of order 1
endomorphism structure, automorphism structure	endomorphism structure of groups of order 1
group cohomology	group cohomology of groups of order 1

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.

Minimal order attaining number

$1$ is the smallest number such that there is precisely $1$ group of that order up to isomorphism. That is, the value of the minimal order attaining function at $1$ is $1$.