# Groups of order 9

From Groupprops

This article gives information about, and links to more details on, groups of order 9

See pages on algebraic structures of order 9| See pages on groups of a particular order

There are, up to isomorphism, *two* possibilities for a group of order 9. Both of these are abelian groups and, in particular are abelian of prime power order.

The classification follows 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:Z9 | 1 | unique cyclic group of order 9 |

elementary abelian group:E9 | 2 | unique elementary abelian group of order 9; also a direct product of two copies of cyclic group:Z3. |