3-step group

In mathematics, a 3-step group is a special sort of group of Fitting length at most 3, that is used in the classification of CN groups and in the Feit–Thompson theorem. The definition of a 3-step group in these two cases is slightly different.

CN groups

In the theory of CN groups, a 3-step group is a group such that:

Any 3-step group is a solvable CN-group, and conversely any solvable CN-group is either nilpotent, or a Frobenius group, or a 3-step group.
Example: the symmetric group S₄ is a 3-step group for the prime.

defined a three-step group to be a group G satisfying the following conditions:

The derived group of G is a Hall subgroup with a cyclic complement Q.
If H is the maximal normal nilpotent Hall subgroup of G, then G⊆''HCG''⊆G and HC_G is nilpotent and H is noncyclic.
For q∈''Q nontrivial, CG'' is cyclic and non-trivial and independent of q.