# Congruence condition

From Groupprops

## Contents

## Definition

A **congruence condition** is a condition on the congruence class, taken modulo some natural number , of some count associated with a group. Typically, we talk of congruence conditions on the order or index of a subgroup, or on a count of the number of subgroups satisfying a particular condition.

Typical congruence conditions are described below:

- modulo for subgroups satisfying certain order conditions and properties: This states that for a given finite group, or a given finite -group, the number of subgroups of order a particular power of dividing the group's order and satisfying some condition is modulo .
- modulo , or zero: This states that for a given finite group, or a given finite -group, the number of subgroups of order a particular power of dividing the group's order and satisfying some condition is modulo . A closely related notion is that of a collection of groups satisfying a universal congruence condition.
- modulo , or a bounded finite number: This is similar to the previous cases, except that we now allow exceptions where the number is finite with a fixed bound. The most typical example is: modulo , or , or .

## Examples

For a complete list, refer Category:Congruence conditions.

### The pure 1 modulo p statements

- Congruence condition on Sylow numbers
- Congruence condition on number of subgroups of given prime power order

### The 1 modulo p or zero statements

- Jonah-Konvisser abelian-to-normal replacement theorem: This states that the number of abelian subgroups of order , for fixed , and odd, is either zero or congruent to modulo .
- Jonah-Konvisser elementary abelian-to-normal replacement theorem
- Congruence condition on number of subgroups of given prime power order and bounded exponent in abelian group
- Jonah-Konvisser congruence condition for abelian maximal subgroups