Groups of order 4

This article gives information about, and links to more details on, groups of order 4
See pages on algebraic structures of order 4 | See pages on groups of a particular order

There are, up to isomorphism, two possibilities for a group of order 4. Both of these are abelian groups and, in particular are abelian of prime power order. 4 is the first natural number such that there are non-isomorphic groups of that order.

The classification can be done by hand using multiplication tables, but it also follows more generally from the classification of groups of prime-square order or the classification of groups of an order two times a prime.

See also groups of prime-square order for side-by-side comparison with the situation for other primes.

The groups are:

Group	GAP ID (second part)	Defining feature
cyclic group:Z4	1	unique cyclic group of order 4
Klein four-group	2	unique elementary abelian group of order 4; also a direct product of two copies of cyclic group:Z2.

Minimal order attaining number

$4$ is the smallest number such that there are precisely $2$ groups of that order up to isomorphism. That is, the value of the minimal order attaining function at $2$ is $4$.