Congruence condition summary for groups of prime-cube order
This article gives specific information, namely, congruence condition summary, about a family of groups, namely: groups of prime-cube order.
View congruence condition summary for group families | View other specific information about groups of prime-cube order
Case of the prime 2
We first provide the congruence condition summary for groups of order 8, i.e., the case where .
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:
Case of the prime 3
We now provide the congruence condition summary for groups of order 27, i.e., the case .
Case of primes greater than 3
For any prime , the congruence condition summary for groups of order
looks the same. It is given below.
First, note that there are five groups of order . The three abelian groups are cyclic group of prime-cube order, direct product of cyclic group of prime-square order and cyclic group of prime order, and elementary abelian group of prime-cube order. The two non-abelian groups are prime-cube order group:U(3,p) (this has exponent
) and semidirect product of cyclic group of prime-square order and cyclic group of prime order (this has exponent
).