The complexity of inversion in groups

We prove that if ${\mathscr A}=(A,\cdot)$ is a group computable in polynomial time (${\rm P}$-computable), then there exists a ${\rm P}$-computable group ${\mathscr B}=(B,\cdot)\cong{\mathscr A}$, in which the operation $x^{-1}$ is also ${\rm P}$-computable. On the other hand, we show that if the center $Z({\mathscr A})$ of a group ${\mathscr A}$ contains an element of infinite order, then under some additional assumptions, there exists a ${\rm P}$-computable group ${\mathscr B}'=(B',\cdot)\cong{\mathscr A}$, in which the operation $x^{-1}$ is not primitive recursive. Also the following general fact in the theory of ${\rm P}$-computable structures is stated: if ${\mathscr A}$ is a ${\rm P}$-computable structure and $E\subseteq A^{2}$ is a ${\rm P}$-computable congruence on ${\mathscr A}$, then the quotient structure ${\mathscr A} / E$ is isomorphic to a ${\rm P}$-computable structure.