  An Eggert-operad is a variant of Mac Lane's notion of a PROP, for which not
only bijective maps, but all maps between standard finite sets, are part of the
structure. We construct the free Eggert-operad and prove the universal property
it satisfies. Further we use the free Eggert-operad to define the Eggert-operad
$\mathrm{GROUP}_0^{\mathrm{fp}}$ via generators and relations and show that a
$\mathrm{GROUP}_0^{\mathrm{fp}}$-algebra is a group and vice versa. At the end
we prove operadic versions of some elementary facts about groups.
