Minimal generating sets in groups of $p$-automata

Abstract

For an arbitrary odd prime $p$, we consider groups of all $p$-automata and all finite $p$-automata. We construct minimal generating sets in both the groups of all $p$-automata and its subgroup of finite $p$-automata. The key ingredient of the proof is the lifting technique, which allows the construction of a minimal generating set in a group provided a minimal generating set in its abelian quotient is given. To find the required quotient, the elements of the groups of $p$-automata and finite $p$-automata are presented in terms of tableaux introduced by L. Kaloujnine. Using this presentation, a natural homomorphism on the additive group of all infinite sequences over the field $\mathbb{Z}_p$ is defined and examined.