Congruence condition
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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