Congruence condition

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

 * 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 $$p^k$$, for fixed $$0 \le k \le 5$$, and $$p$$ odd, is either zero or congruent to $$1$$ modulo $$p$$.
 * 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