Three-step group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This term was introduced by: Georenstein

Definition

A finite group $G$ is said to be a three-step group if it satisfies all the following conditions:

• The commutator subgroup $G'$ has a permutable complement $Q$ which is a cyclic Hall subgroup
• $G''\leq F(G)\leq G'$ where $F(G)$ is the Fitting subgroup of $G$
• If $H$ is a largest Hall subgroup of $G$ contained in $F(G)$, then $H$ is non-cyclic and $F(G)=HC_{G}(H)$
• $H$ contains a cyclic subgroup $R$ such that $C_{G'}(y)=R$ for all non-identity elements $y$ of $H$.