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.

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 81b:20002