Normality and automata

We prove that finite-state transducers with injective behavior, deterministic or not, real-time or not, with no extra memory or a single counter, cannot compress any normal word. We exhaust all combinations of determinism, real-time, and additional memory in the form of counters or stacks, identifying which models can compress normal words. The case of deterministic push-down transducers is the only one still open. We also present results on the preservation of normality by selection with finite automata. Complementing Agafonov's theorem for prefix selection, we show that suffix selection preserves normality. However, there are simple two-sided selection rules that do not.