3-step group

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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[edit]

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[edit]

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