  A group of permutations G of a set V is k-distinguishable if there exists a
partition of V into k parts such that only the identity permutation in G fixes
setwise all of the cells of the partition. The least cardinal number k such
that (G,V) is k-distinguishable is its distinguishing number. In particular, a
graph X is k-distinguishable if its automorphism group Aut(X) has
distinguishing number at most k in its action on the vertices of X.
  Various results in the literature demonstrate that when an infinite graph
fails to have some property, then often some finite subgraph is similarly
deficient. In this paper we show that whenever an infinite connected graph X is
not k-distinguishable (for a given cardinal k), then it contains a ball B of
finite radius whose distinguishing number is at least k. Moreover, this lower
bound cannot be sharpened, since for any integer k greater than 3 there exists
an infinite, locally finite, connected graph X that is not k-distinguishable
but in which every ball of finite radius is k-distinguishable.
  In the second half of this paper we show that a large distinguishing number
for an imprimitive graph X is traceable to a high distinguishing number either
of a block of imprimitivity or of the induced action of Aut(X) on the
corresponding system of imprimitivity. The distinguishing numbers of infinite
primitive graphs have been examined in detail in a previous paper by the
authors together with Tom W. Tucker.
