Group whose Frattini series equals its derived series

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


 * It is a defining ingredient::solvable group, so its defining ingredient::derived series terminates in finitely many steps at the trivial subgroup.
 * Its defining ingredient::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.

Stronger properties

 * Weaker than::Elementary Abelian group
 * Weaker than::Special group
 * Weaker than::Extraspecial group

Related properties

 * Group whose lower central series factor groups are elementary Abelian