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A group of permutations G of a set V is k-distinguishable if there exists apartition of V into k parts such that only the identity permutation in G fixessetwise all of the cells of the partition. The least cardinal number k suchthat (G,V) is k-distinguishable is its distinguishing number. In particular, agraph X is k-distinguishable if its automorphism group Aut(X) hasdistinguishing number at most k in its action on the vertices of X.  Various results in the literature demonstrate that when an infinite graphfails to have some property, then often some finite subgraph is similarlydeficient. In this paper we show that whenever an infinite connected graph X isnot k-distinguishable (for a given cardinal k), then it contains a ball B offinite radius whose distinguishing number is at least k. Moreover, this lowerbound cannot be sharpened, since for any integer k greater than 3 there existsan infinite, locally finite, connected graph X that is not k-distinguishablebut in which every ball of finite radius is k-distinguishable.  In the second half of this paper we show that a large distinguishing numberfor an imprimitive graph X is traceable to a high distinguishing number eitherof a block of imprimitivity or of the induced action of Aut(X) on thecorresponding system of imprimitivity. The distinguishing numbers of infiniteprimitive graphs have been examined in detail in a previous paper by theauthors together with Tom W. Tucker.

Bounding the distinguishing number of infinite graphs