  We give new characterizations of sofic groups:
  -- A group $G$ is sofic if and only if it is a subgroup of a quotient of a
direct product of alternating or symmetric groups.
  -- A group $G$ is sofic if and only if any system of equations solvable in
all alternating groups is solvable over $G$.
  The last characterization allows to express soficity of an existentially
closed group by $\forall\exists$-sentences.
  Keywords: sofic groups, approximations, equations over groups.
