Maximal subgroups of finite soluble groups in general position

Abstract

For a finite group G we investigate the difference between the maximum size $${{\mathrm{MaxDim}}}(G)$$ of an “independent” family of maximal subgroups of G and maximum size m(G) of an irredundant sequence of generators of G. We prove that $${{\mathrm{MaxDim}}}(G)=m(G)$$ if the derived subgroup of G is nilpotent. However, $${{\mathrm{MaxDim}}}(G)-m(G)$$ can be arbitrarily large: for any odd prime p, we construct a finite soluble group with Fitting length two satisfying $$m(G)=3$$ and $${{\mathrm{MaxDim}}}(G)=p$$ .