  We prove that the set of subgroups of the automorphism group of a two-sided
full shift is closed under graph products. We introduce the notion of an
Aithful group action, and show that when A is a finite abelian group and G is a
group of cellular automata whose action is Aithful, the wreath product A \wr G
embeds in the automorphism group of a full shift. We show that all free abelian
groups and free groups admit Aithful cellular automata actions. In the
one-sided case, we prove variants of these result with reasonable alphabet
blow-ups.
