Congruence condition

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Definition

A congruence condition is a condition on the congruence class, taken modulo some natural number n, 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:

  1. 1 modulo p for subgroups satisfying certain order conditions and properties: This states that for a given finite group, or a given finite p-group, the number of subgroups of order a particular power of p dividing the group's order and satisfying some condition is 1 modulo p.
  2. 1 modulo p, or zero: This states that for a given finite group, or a given finite p-group, the number of subgroups of order a particular power of p dividing the group's order and satisfying some condition is 1 modulo p. A closely related notion is that of a collection of groups satisfying a universal congruence condition.
  3. 1 modulo p, 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: 1 modulo p, or 0, or 2.

Examples

For a complete list, refer Category:Congruence conditions.

The pure 1 modulo p statements

The 1 modulo p or zero statements