# Collection of groups satisfying a universal congruence condition

From Groupprops

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## Definition

Suppose is a finite collection of finite -groups, groups of prime power order for the prime . We say that satisfies a universal congruence condition if the following equivalent conditions are satisfied by :

- For any finite -group that contains a subgroup isomorphic to an element of , the number of subgroups of isomorphic to elements of is congruent to modulo .
- For any finite -group that contains a subgroup isomorphic to an element of , the number of normal subgroups of isomorphic to elements of is congruent to modulo .
- For any finite -group and any normal subgroup of such that contains a subgroup isomorphic to an element of , the number of normal subgroups of isomorphic to elements of and contained in is congruent to modulo .
- For any finite -group that contains a subgroup isomorphic to an element of , the number of p-core-automorphism-invariant subgroups of isomorphic to elements of is congruent to modulo .
- For any finite group containing a subgroup isomorphic to an element of , the number of subgroups of isomorphic to an element of is congruent to modulo .

### Equivalence of definitions

`Further information: equivalence of definitions of universal congruence condition`

## Relation with other properties

### Weaker properties

- Collection of groups satisfying a strong normal replacement condition
- Collection of groups satisfying a weak normal replacement condition

## Examples/facts

### Satisfaction

### Dissatisfaction

Collection | Conditions on prime | Conditions on | Proof |
---|---|---|---|

Klein four-group | elementary abelian-to-normal replacement fails for Klein four-group | ||

Elementary abelian group of order | Follows from above | ||

Elementary abelian group of order | all | ||

Abelian groups of order | all | Congruence condition fails for abelian subgroups of prime-sixth order | |

Abelian groups of order | all | Follows from above. | |

Elementary abelian group of order | all | Congruence condition fails for elementary abelian subgroups of prime-sixth order (example same as for abelian subgroups) | |

Elementary abelian group of order | all | Follows from above. | |

Groups of order , exponent | all | Congruence condition fails for subgroups of order p^p and exponent p |

### Threshold values

This lists threshold values of : the largest value of for which the collection of -groups of order satisfying the stated condition satisfies a universal congruence condition. The nature of all these is such that the universal congruence condition is satisfied for all smaller but for no larger . We use *between and * to mean that the value is at least and at most .

Collection of groups | |||||
---|---|---|---|---|---|

Abelian groups of order | between 4 and 5 | 5 | 5 | 5 | 5 |

Abelian groups of order , exponent dividing , | between 3 and 5 | 5 | 5 | 5 | 5 |

Elementary abelian group of order | 1 | 5 | 5 | 5 | 5 |

Groups of exponent , order | 1 | 2 | between 2 and 4 | between 2 and 6 | between 2 and |