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:

* G = Op,p',p(G)
* Op,p′(G) is a Frobenius group with kernel Op(G)
* G/Op(G) is a Frobenius group with kernel Op,p′(G)/Op(G)

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) 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(H)⊆G′ and HCG is nilpotent and H is noncyclic.
* For q∈Q nontrivial, CG(q) is cyclic and non-trivial and independent of q.

## References

* Feit, Walter; Thompson, John G. (1963), "Solvability of groups of odd order", Pacific Journal of Mathematics 13: 775–1029, doi:10.2140/pjm.1963.13.775, ISSN 0030-8730, http://projecteuclid.org/Dienst/UI/1.0/Journal?authority=euclid.pjm&issue=1103053941
* Feit, Walter; Thompson, John G.; Hall, Marshall Jr (1960), "Finite groups in which the centralizer of any non-identity element is nilpotent", Mathematische Zeitschrift 74: 1–17, doi:10.1007/BF01180468, ISSN 0025-5874
* Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6

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