In this paper, all groups and graphs considered are finite and all graphs are simple (in the sense of Tutte [8, p. 50]). If X is such a graph with vertex set V(X) and automorphism group A(X), we say that X is a graphical regular representation (GRR) of a given abstract group G if (I) G ^ ^ ( Z ) , a n d (II) A(X) acts on V(X) as a regular permutation group; that is, given u, v G V(X), there exists a unique <p £ A (X) for which <p(u) = v. That for any abstract group G there exists a graph X satisfying (I) is well-known (cf. [3]). The question of existence or non-existence of a GRR for a given abstract group G, however, has been settled to date only for relatively few classes of groups. This problem is the underlying motivation for this paper and its sequel. In Section 1 we introduce some essential notation and attempt a summary of what is known to date (with references to the literature) about the foregoing problem. In Section 2 some machinery involving at one time techniques from both group theory and graph theory is developed in order to facilitate proving, when true, that a given graph is indeed a GRR. These techniques are then used to prove the main result of the section:
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