16.45 (2006)

Let $G$ be a permutation group on a set $\Omega$. A sequence of points of $\Omega$ is a base for $G$ if its pointwise stabilizer in $G$ is the identity; it is minimal if no point may be removed. Let $b(G)$ be the maximum, over all permutation representations of the finite group $G$, of the maximum size of a minimal base for $G$. Let $\mu'(G)$ be the maximum size of an independent set in $G$, a set of elements with the property that no element belongs to the subgroup generated by the others. Is it true that $b(G) = \mu'(G)$? (It is known that $b(G) \leqslant \mu'(G)$, and that equality holds for the symmetric groups.)

Remark. An equivalent question is the following. Suppose that the Boolean lattice $B(n)$ of subsets of an $n$-element set is embeddable as a meet-semilattice of the subgroup lattice of $G$, and suppose that $n$ is maximal with this property. Is it true that then there is such an embedding of $B(n)$ with the property that the least element of $B(n)$ is a normal subgroup of $G$?