  Clozel, Harris and Taylor have recently proved a modularity lifting theorem
of the following general form: if rho is an l-adic representation of the
absolute Galois group of a number field for which the residual representation
rho-bar comes from a modular form then so does rho. This theorem has numerous
hypotheses; a crucial one is that the image of rho-bar must be "big," a
technical condition on subgroups of GL(n). In this paper we investigate this
condition in compatible systems. Our main result is that in a sufficiently
irreducible compatible system the residual images are big at a density one set
of primes. This result should make some of the work of Clozel, Harris and
Taylor easier to apply in the setting of compatible systems.
