# P-central 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

## Contents

## Definition

### For the case of an odd prime

Let be an odd prime. A **p-central group** is a p-group (i.e., a group in which the order of every element is a power of a fixed prime number ) with the property that all the elements of order are inside the center. In the finite case, this is equivalent to saying that the first omega subgroup is contained in the center .

### For the case

For the prime , a **p-central group** (or a -central group in this case) is a p-group (i.e., a group in which the order of every element is a power of a fixed prime number ) with the property that all elements of order or are in the center. In the finite case, this is equivalent to saying that the second omega subgroup is contained in the center .