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 group CN Group Limited was formerly an independent local media business based in Carlisle, Cumbria, England, operating in print and radio. It is now owned by Newsquest and their newspapers are printed in Glasgow. The company was formerly known as t ...

s and in the

Feit–Thompson theorem In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by . History conjectured that every nonabelian finite simple group has even order. suggested using ...

. 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: * * is a Frobenius group with kernel * is a Frobenius group with kernel 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 .

Odd order groups

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

* * *{{Citation , last1=Gorenstein , first1=D. , author1-link=Daniel Gorenstein , title=Finite Groups , publisher=Chelsea , location=New York , isbn=978-0-8284-0301-6 , mr=569209 , year=1980 Finite groups