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 (for some prime p) 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 p=2.

Odd order groups

Feit & Thompson (1963, p.780) defined a three-step group to be a group G satisfying the following conditions:

References

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