# Congruence condition summary for groups of order 3^n

This article gives specific information, namely, congruence condition summary, about a family of groups, namely: groups of order 3^n.

View congruence condition summary for group families | View other specific information about groups of order 3^n

This article gives relevant information on:

- Every possible collection of groups satisfying a universal congruence condition relative to the prime 3, for small orders.
- Conditional versions of congruence conditions, e.g., those obtained by restricting the size or putting some other constraint on the size of the ambient group.

## Universal and conditional congruence conditions by order

### Order 1

The only nontrivial collection of groups of order 1 is the singleton collection comprising the trivial group, and this satisfies a universal congruence condition *and* an existence condition.

### Order 3

The only nontrivial collection of groups of order 1 is the singleton collection comprising cyclic group:Z3, and this satisfies a universal congruence condition *and* an existence condition.

### Order 9

There are two groups of order 9: cyclic group:Z9 and elementary abelian group:E9. There are thus possible non-empty collections of groups of this order. We note which of these satisfy the congruence condition:

Collection | Does it satisfy a universal congruence condition? | Restricted class of groups in which it satisfies a congruence condition | Explanation |
---|---|---|---|

cyclic group:Z9 only | No | cyclic groups | |

elementary abelian group:E9 only | Yes | all groups | congruence condition on number of elementary abelian subgroups of prime-square order for odd prime |

cyclic group:Z9 and elementary abelian group:E9 | Yes | all groups | congruence condition on number of subgroups of given prime power order |

### Order 27

There are five groups of order 27. The three abelian groups are cyclic group:Z27, direct product of Z9 and Z3, and elementary abelian group:E27. The two non-abelian groups are prime-cube order group:U(3,3) (which has exponent 3) and M27 (which has exponent 9, and is a semidirect product of a cyclic group of order nine by a cyclic group of order three).

There are thus possible collections of groups. Instead of listing all 31, we simply note the ones that *do* satisfy a universal congruence condition or a congruence condition to an interesting restricted class: