On the minimal number of generators of endomorphism monoids of full shifts.

For a group G and a finite set A, denote by End (A G) the monoid of all continuous shift commuting self-maps of A G and by Aut (A G) its group of units. We study the minimal cardinality of a generating set, known as the rank, of End (A G) and Aut (A G) . In the first part, when G is a finite group, we give upper and lower bounds for the rank of Aut (A G) in terms of the number of conjugacy classes of subgroups of G. In the second part, we apply our bounds to show that if G has an infinite descending chain of normal subgroups of finite index, then End (A G) is not finitely generated; such is the case for wide classes of infinite groups, such as infinite residually finite or infinite locally graded groups. [ABSTRACT FROM AUTHOR]