The Schur Group of a Closed Subset

Recall that the group of all bijective faithful maps from xT to xT is called the Schur group of T with respect to x. The identity on xT is the identity element of the Schur group of T with respect to x. 

Recall that a closed subset of S is called schurian if it is faithfully embedded in itself. 

Theorem 6.3.1 Let x be an element in X, and let T be a closed subset of S. Then the following statements are equivalent. 

Let w be an element in xT, and let t be an element in T such that w G xt. Since G is assumed to act transitively on xT, we shall be done if we succeed in showing that w f G xftxTf. 

Since G acts transitively on xT, there exists an element g in G such that w = xg 1. Thus, by definition, uxp = gHT. 

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Theorem 6.3.3 A closed subset T of S is schurian if and only if for each element x in X, the Schur group G of T with respect to x acts transitively on xT and, for each element t in T, Gx acts transitively on xt. 

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