Cayley's Theorem for Topological Groups

This result appears in every presentation of the elements of group theory (see, e.g., [10], [2], [7], and [6]). It has been generalized to infinite groups by replacing S, by the group of transformations on a set of appropriate cardinality, and analogs of (*) for monoids, rings, and algebras appear in [8]. The qualifier "almost" appears in the first sentence above because of Burnside's attribution (in [4]) of (*) to Jordan. The collected works of Cayley are readily accessible. They indicate that in 1854 Cayley defined the notion of (finite) group. He emphasized the abstract nature of a group by asserting that "a group is defined by means of the laws of combination of its symbols." He also represented a group by its multiplication table (sometimes called a Cayley table), and remarked that each row (column) of the table contains all the group elements, each one appearing only once [5]. Thus he implicitly makes the connection between group elements and regular permutations, but he does not explicitly prove (*). Moreover, as Burnside quite correctly notes, Jordan did indeed prove (*) in 1870 [9, pp. 60-61]. Nevertheless, to call (*) Cayley's Theorem is not an inappropriate attribution. Cayley essentially showed that the correspondence which takes a group element a to the permutation Ta (where T,(x)= ax for any group element x) is one-to-one, even though he failed to demonstrate explicitly that T'aTb=T,Tb. More important, he communicated his awareness of this correspondence to the mathematical community at large. Having introduced Cayley's Theorem, we wish to observe that it has a natural generalization to topological groups. In this case, sets are replaced by uniform spaces and the group of permutations of a set by the topological automorphism group of a uniform space. Before stating (and proving) the topological Cayley Theorem, we record the relevant definitions (as found in, e.g., [3]). If X is a uniform space, then its automorphism group Aut(X) consists of all uniformly continuous functions X-*X with uniformly continuous inverse, the topology being that of uniform convergence. For this topology, a fundamental system of neighborhoods of the identity automorphism is given by the sets 'V = {(al(x, a(x)) e V for all x E X }, where V ranges over the entourages of the uniformity of X. It is routine to verify that this fundamental system of neighborhoods of the identity is compatible with the group structure of Aut(X), and thus that Aut(X) is a topological group. Moreover, if X is Hausdorff (respectively, discrete), then Aut(X) is Hausdorff (respectively, discrete). These last two observations emphasize that our topological discussion is not limited to Hausdorff topological groups only, and that it is really a generalization of the algebraic situation. Now if V is a neighborhood of the identity e of the topological group G, we set Vd = {(x,y) E G X G Iyx -E V). Then as V runs through a fundamental system of neighborhoods of e, the sets Vd describe a fundamental system of entourages of ,t, the right uniformity of G. (The subscript d stands for "dextra," the Latin word for "right." If we had multiplied y on the left by x -, we would have obtained the left uniformity on G, which is in general distinct from the right uniformity. In this case we would have used a subscript s standing for "sinistra.") We can now state the analog of Cayley's Theorem for topological groups: