Automorphisms of K are bijective and so belong to Sym(K).
The subset of all automorphisms is not empty as the identity is an automorphism.
If g is an automorphism of K, then so is g-1 (by definition if K is a graph, by a previous theorem if K is a ring).
Likewise, if g and h
are automorphisms of
K, then so is the composition g ·
h.