This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
- For every , there exists such that .
- For every , there exist such that .
- For every and every natural number , there exist such that where .
The term regular p-group is typically used only for finite p-groups.
Relation with other properties
|Metaproperty name||Satisfied?||Proof||Statement with symbols|
|subgroup-closed group property||Yes||regular p-group property is subgroup-closed||If is prime, is a regular -group, and is a subgroup of , then is a regular -group.|
|quotient-closed group property||Yes||regular p-group property is quotient-closed||If is prime, is a regular -group, is a normal subgroup of , and is the corresponding quotient group, then is also a regular -group.|
|finite direct product-closed group property||No||regular p-group property is finite direct product-closed||It is possible to have a prime number and regular -groups such that the external direct product is not a regular -group.|
|2-local group property||Yes||regular p-group property is 2-local||Suppose is a prime number and is a group such that for all , is a regular -group, then itself is a regular -group.|