# Groups of order 4

From Groupprops

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.

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.

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. |