# 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*= O_{p,p′,p}(*G*)- O
_{p,p′}(*G*) is a Frobenius group with kernel O_{p}(*G*) *G*/O_{p}(*G*) is a Frobenius group with kernel O_{p,p′}(*G*)/O_{p}(*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 *S*_{4} 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:

- 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*′′⊆*H*C_{G}(*H*)⊆*G*′ and*H*C_{G}is nilpotent and*H*is noncyclic. - For
*q*∈*Q*nontrivial, C_{G}(*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, MR 0166261 - 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, MR 0114856 - Gorenstein, D. (1980),
*Finite Groups*, New York: Chelsea, ISBN 978-0-8284-0301-6, MR 0569209

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