A common technique of study is to focus on a subset of the properties or a portion of the object under consideration; by understanding a small piece, one may be able to say something about the whole, at least at a local level.  So for studying commutativity, it helps to pick two arbitrary elements and look at the (subgroup, except I will switch to universal algebra mode now, and say) subalgebra they generate.  Similarly, using congruence conditions modulo various primes, one can say some things about the solutions to various Diophantine equations.

A theorem in general algebra is Garrett Birkhoff's Subdirect Representation theorem: any algebraic structure is a subdirect product of subdirectly irreducible algebras. (See e.g. Algebras, Lattices, Varieties by McKenzie, McNulty, Taylor for definitions and a precise statement.)  This gives us a possible tool for study: for the group or ring or cylindric algebra of interest, project it via a homomorphism onto a small subdirectly irreducible component, and see what can be said on that component. 

For me, the big picture is what analytical tools I can use.  For you, the big picture may be how the large structure is put together.  Residual finiteness, like subdirect representation, is a tool that you can use to determine whether and how examining the algebra locally will help.

Gerhard "Remember Picture In Picture Feature" Paseman, 2015.02.26