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.
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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
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.
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.
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|
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: