  We discuss the set of subgroups of the automorphism group of a full shift,
and submonoids of its endomorphism monoid. We prove closure under direct
products in the monoid case, and free products in the group case. We also show
that the automorphism group of a full shift embeds in that of an uncountable
sofic shift. Some undecidability results are obtained as corollaries.

Dynamical Systems;Combinatorics;Group Theory