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 (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 GHCG(H)⊆G and HCG is nilpotent and H is noncyclic.
  • For qQ nontrivial, CG(q) is cyclic and non-trivial and independent of q.


  • 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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