TITLE: Automorphisms of a structure as a powerful tool for studying the structure
QUESTION [5 upvotes]: This is just an arbitrary testimony of an often repeated slogan:

"The group of automorphisms of a given structure is often a powerful
  tool for studying this structure." D. Lascar, On the Category of
  Models of a Complete Theory

I wonder why this should be so when thinking of the fact that (for example) almost all graphs have a trivial automorphism group. And only very few - if any - graphs have an automorphism group that allows literally to tell the graph from it.

How then is this slogan to be understood? For which kinds of structures does it make sense, resp. in which sense?

Side question: Is it true, that when the automorphism group of a graph is isomorphic to the full symmetric group $S_n$, then the graph has to be the complete graph $K_n$ or its complement?

REPLY [3 votes]: I wonder why this should be so when thinking of the fact that (for example) almost all graphs have a trivial automorphism group.

The distribution of graphs that one can encounter is biased in favor of nonrandom, structured, low entropy objects.