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For a group $G$ and a finite set $A$, denote by $\text{CA}(G;A)$ the monoidof all cellular automata over $A^G$ and by $\text{ICA}(G;A)$ its group ofunits. We study the minimal cardinality of a generating set, known as the rank,of $\text{ICA}(G;A)$. In the first part, when $G$ is a finite group, we giveupper bounds for the rank in terms of the number of conjugacy classes ofsubgroups of $G$. The case when $G$ is a finite cyclic group has been studiedbefore, so here we focus on the cases when $G$ is a finite dihedral group or afinite Dedekind group. In the second part, we find a basic lower bound for therank of $\text{ICA}(G;A)$ when $G$ is a finite group, and we apply this to showthat, for any infinite abelian group $H$, the monoid $\text{CA}(H;A)$ is notfinitely generated. The same is true for various kinds of infinite groups, sowe ask if there exists an infinite group $H$ such that $\text{CA}(H;A)$ isfinitely generated.

Bounding the minimal number of generators of groups and monoids of cellular automata