  We say that a finite group $G$ satisfies the independence property if, for
every pair of distinct elements $x$ and $y$ of $G$, either $\{x,y\}$ is
contained in a minimal generating set for $G$ or one of $x$ and $y$ is a power
of the other. We give a complete classification of the finite groups with this
property, and in particular prove that every such group is supersoluble. A key
ingredient of our proof is a theorem showing that all but three finite almost
simple groups $H$ contain an element $s$ such that the maximal subgroups of $H$
containing $s$, but not containing the socle of $H$, are pairwise
non-conjugate.
