# 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: