# Group whose Frattini series equals its derived series

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

## Definition

A group whose Frattini series equals its derived series is a group with the following three properties:

• It is a solvable group, so its derived series terminates in finitely many steps at the trivial subgroup.
• Its Frattini series terminates in finitely many steps at the trivial subgroup.
• The Frattini series and derived series are identical.

Note that since, for a finite group, the Frattini subgroup is nilpotent, any group for which the Frattini subgroup equals the commutator subgroup must be solvable, so for finite groups, some of these conditions may be redundant.

Equivalently, a group whose Frattini series equals its derived series is a solvable group such that all the factor groups of its derived series are Frattini-free groups. In the case of a group of prime power order, this is equivalent to saying that all the quotients are elementary Abelian groups.