15.45 (2002)

We define the class of hierarchically decomposable groups in the following way. First, if $\mathfrak{X}$ is any class of groups, then let $\mathbf{H}_1 \mathfrak{X}$ denote the class of groups which admit an admissible action on a finite-dimensional contractible complex in such a way that every cell stabilizer belongs to $\mathfrak{X}$. Then the ``big'' class $\mathbf{H}\mathfrak{X}$ is defined to be the smallest $\mathbf{H}_1$-closed class containing $\mathfrak{X}$.
$\qquad$ a) Let $\mathfrak{F}$ be the class of all finite groups. Find an example of an $\mathbf{H}\mathfrak{F}$ group which is not in $\mathbf{H}_3\mathfrak{F}$ ($= \mathbf{H}_1\mathbf{H}_1\mathbf{H}_1\mathfrak{F}$).
$\qquad$ b) Prove or disprove that there is an ordinal $\alpha$ such that $\mathbf{H}_\alpha\mathfrak{F} = \mathbf{H}\mathfrak{F}$, where $\mathbf{H}_\alpha$ is the operator on classes of groups defined by transfinite induction in the obvious way starting from $\mathbf{H}_1$.