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
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:Z2, and this satisfies a universal congruence condition and an existence 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|
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: