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Hua's fundamental theorem of geometry of hermitian matrices characterizes all bijective maps on the space of all hermitian matrices, which preserve adjacency in both directions. In this and subsequent paper we characterize maps on the set of all invertible hermitian matrices over a finite field, which preserve adjacency in one direction. This and author's previous result are used to obtain two new results related to maps that preserve the `speed of light' on finite Minkowski space-time. In this first paper it is shown that maps that preserve adjacency on the set of all invertible hermitian matrices over a finite field are necessarily bijective, so the corresponding graph on invertible hermitian matrices, where edges are defined by the adjacency relation, is a core. Besides matrix theory, the proof relies on results from several other mathematical areas, including spectral and chromatic graph theory, and projective geometry

Maps that preserve adjacency on the set of all invertible hermitian matrices over a finite field are characterized. It is shown that such maps form a group that is generated by the maps $A\mapsto PAP^{\ast}$, $A\mapsto A^{\sigma}$, and $A\mapsto A^{-1}$, where $P$ is an invertible matrix, $P^{\ast}$ is its conjugate transpose, and $\sigma$ is an automorphism of the underlying field. Bijectivity of maps is not an assumption but a conclusion. Moreover, adjacency is assumed to be preserved in one directions only. The main result and author's previous result [16] are applied to characterize maps that preserve the `speed of light' on (a) finite Minkowski space-time and (b) the complement of the light cone in finite Minkowski space-time

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