# Powerful 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

A powerful p-group is a p-group satisfying the condition below. The term is typically used only for finite p-groups, but the definition makes sense in the infinite context as well.

### For the prime

A 2-group (i.e., a finite group of order a power of 2) is termed **powerful** if where is the derived subgroup and is the second agemo subgroup, i.e., the subgroup generated by the powers of elements.

### For odd primes

Suppose is a -group, an odd prime. is termed **powerful** if where is the derived subgroup and is the first agemo subgroup, i.e., the subgroup generated by all powers.