Congruence condition summary for groups of order 3^n

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This article gives specific information, namely, congruence condition summary, about a family of groups, namely: groups of order 3^n.
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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 3

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.

Order 9

There are two groups of order 9: cyclic group:Z9 and elementary abelian group:E9. There are thus 2^2 - 1 = 3 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

Order 27

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 2^5 - 1 = 31 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:

Collection Does it satisfy a universal congruence condition? Restricted class of groups in which it satisfies a congruence condition Explanation
elementary abelian group:E27 only Yes all groups congruence condition on number of elementary abelian subgroups of prime-cube and prime-fourth order
direct product of Z9 and Z3 and elementary abelian group:E27 Yes all groups Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order and bounded exponent for odd prime
cyclic group:Z27, direct product of Z9 and Z3, elementary abelian group:E27 Yes all groups Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime
direct product of Z9 and Z3, prime-cube order group:U(3,3), M27, elementary abelian group:E27 Yes all groups congruence condition on number of non-cyclic subgroups of prime-cube order for odd prime
all groups of order 27 Yes all groups congruence condition on number of subgroups of given prime power order
prime-cube order group:U(3,3) and elementary abelian group:E27 No groups of nilpotency class two congruence condition on number of subgroups of given order and bounded exponent in class two group for odd prime
Why it's not true in general: Congruence condition fails for subgroups of order p^p and exponent p. In general, we can get a failure inside wreath product of Z3 and Z3 (order 81).

Order 81