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