Finite Unitary Geometry and PBIB Designs \((I)\)

The conjugation relation among the subspaces of a finite unitary geometry and its properties are studied. Then they are used to find some enumeration formulas for the subspaces of the unitary geometry, to prove a type of transitivity of the unitary group, to construct Partially Balanced Incomplete Block (PBIB) designs, and to establish the isomorphism between some known PBIB designs.