Congruence condition summary for groups of order 2^n
This article gives specific information, namely, congruence condition summary, about a family of groups, namely: groups of order 2^n.
View congruence condition summary for group families | View other specific information about groups of order 2^n
This article gives relevant information on
- Every possible collection of groups satisfying a universal congruence condition relative to the prime 2, for small orders.
- Conditional versions of congruence conditions, e.g., those obtained by restricting the size or putting some other constraint on 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 2
The only nontrivial collection of groups of order 1 is the singleton collection comprising cyclic group:Z2, and this satisfies a universal congruence condition and an existence condition.
Order 4
There are two groups of order 4: cyclic group:Z4 and Klein four-group. There are thus 
possible collections of groups. 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:Z4 only | No | Nothing interesting (?) | Doesn't even satisfy a congruence condition in abelian groups, such as direct product of Z4 and Z2. |
| Klein four-group only | No | abelian groups | congruence condition on number of subgroups of given prime power order and bounded exponent in abelian group explains why it is true in abelian groups. The smallest failure among non-abelian groups occurs in dihedral group:D8. |
| cyclic group:Z4 and Klein four-group | Yes | all groups | congruence condition on number of subgroups of given prime power order |
Order 8
There are five groups of order 8: cyclic group:Z8, direct product of Z4 and Z2, dihedral group:D8, quaternion group, and elementary abelian group:E8. 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:
