Note on invariance and finiteness for the exponent of Hopf algebras

There are two notions of exponent of finite-dimensional Hopf algebras introduced and studied in the literature. In this note, we discuss and compare their properties including invariance and finiteness. Specifically, one notion is invariant under twisting and taking the Drinfeld double, exactly as the other one. We also find that if the non-cosemisimplicity and dual Chevalley property hold, both exponents are infinite in characteristic 0 but finite in positive characteristic.