3-step group

CN groups

• G = O_p,p′,p
• O_p,p′ is a Frobenius group with kernel O_p
• G/O_p is a Frobenius group with kernel O_p,p′/O_p

Odd order groups

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