Introduction

All groups considered in this paper are finite.

In Zur Theorie der endlichen aüflosbaren Gruppen [10] Gaschütz defined a formation as a class of groups ℱwith the following properties:

1) every homomorphic image of an ℱ–group is an ℱ–group.

2) If G/M and G/N are ℱ-groups then G/M ∩ N is an ℱ-group.

In [10] a formation ℱ is said to be saturated if the group G belongs to ℱ whenever G/Φ(G) is in ℱ. In the universe of finite solvable groups the definition of a saturated formation provided a context in which to study extensions of the properties of Carter subgroups to classes other than the nilpotent groups. The importance of saturated formations and their relationship to ℱ -covering subgroups is well known. Zur Theorie der endlichen auflösbaren Gruppen also provided the spur for other developments in finite group theory. In [10] Gaschütz proved that if ℱ is a saturated formation that contains the nilpotent groups, then ℱ has the following property:

For H ⊴ G, if H/H ∩ Φ(G) belongs to ℱ then H belongs to ℱ.

Note that any formation ℱ that has the property (*) must necessarily be saturated and contain all groups of prime order.