# Congruence condition summary for groups of order 5^n

## Contents

This article gives specific information, namely, congruence condition summary, about a family of groups, namely: groups of order 5^n.
View congruence condition summary for group families | View other specific information about groups of order 5^n

## 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 5

The only nontrivial collection of groups of order 1 is the singleton collection comprising cyclic group:Z5, and this satisfies a universal congruence condition and an existence condition.

### Order 25

There are two groups of order 25: cyclic group:Z25 and elementary abelian group:E25. 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:Z25 only No cyclic groups
elementary abelian group:E25 only Yes all groups congruence condition on number of elementary abelian subgroups of prime-square order for odd prime
cyclic group:Z25 and elementary abelian group:E25 Yes all groups congruence condition on number of subgroups of given prime power order

### Order 125

There are five groups of order 125. The three abelian groups are cyclic group:Z125, direct product of Z27 and Z3, and elementary abelian group:E125. The two non-abelian groups are prime-cube order group:U(3,5) (which has exponent 5) and M125 (which has exponent 25, 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:E125 only Yes all groups congruence condition on number of elementary abelian subgroups of prime-cube and prime-fourth order
direct product of Z25 and Z5 and elementary abelian group:E125 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:Z125, direct product of Z25 and Z5, elementary abelian group:E125 Yes all groups Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime
direct product of Z25 and Z5, prime-cube order group:U(3,5), M125, elementary abelian group:E125 Yes all groups congruence condition on number of non-cyclic subgroups of prime-cube order for odd prime
all groups of order 125 Yes all groups congruence condition on number of subgroups of given prime power order
prime-cube order group:U(3,5) and elementary abelian group:E125 Yes all groups Mann's replacement theorem for subgroups of prime exponent