AUTOMORPHISM-GROUPS OF COMPARABILITY AND COVERING GRAPHS

Let G be a group and H a subgroup of G. It is shown that there exists a partially ordered set (X, less-than-or-equal-to) such that G is isomorphic to the group of all automorphisms of the comparability graph of (X, less-than-or-equal-to) and such that under this isomorphism H is mapped onto the group of all order-automorphisms of (X, less-than-or-equal-to). There also exists a partially ordered set (Y, less-than-or-equal-to) such that G is isomorphic to the group of all automorphisms of the covering graph of (Y, less-than-or-equal-to) and such that under this isomorphism H is mapped onto the group of all order-automorphisms of (Y, less-than-or-equal-to). In this representation X and Y can be taken to be finite if G is finite and of the same cardinality as G if G is infinite.