# Regular p-group

From Groupprops

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 propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

Suppose is a prime number. A p-group (i.e., a group where the order of every element is a power of ) is termed a **regular -group** if it satisfies the following equivalent conditions:

- 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

### Stronger properties

## Metaproperties

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. |