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 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 G′′⊆HC_G(H)⊆G′ and HC_G is nilpotent and H is noncyclic.
• For q∈Q nontrivial, C_G(q) is cyclic and non-trivial and independent of q.