Congruence condition summary for 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.

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 $$2^2 - 1 = 3$$ possible non-empty collections of groups of this order. We note which of these satisfy the congruence condition:

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 $$2^5 - 1 = 31$$ 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: