The Complexity of Inversion in Groups

Abstract

We prove that if \(\mathcal{A}\) = (A,⋅) is a group computable in polynomial time (P-computable), then there exists a P-computable group \(\mathcal{B}\) = (B,∙) ≅ \(\mathcal{A},\) in which the operation x⁻¹ is also P-computable. On the other hand, we show that if the center \(Z\left(\mathcal{A}\right)\) of a group A contains an element of infinite order, then under some additional assumptions, there exists a P-computable group \({\mathcal{B}}{\prime}=\left({B}{\prime},\cdot \right)\cong \mathcal{A}\) in which the operation x⁻¹ is not primitive recursive. Also the following general fact in the theory of P-computable structures is stated: if \(\mathcal{A}\) is a P-computable structure and E ⊆ A² is a P-computable congruence on \(\mathcal{A},\) then the quotient structure \(\mathcal{A}/E\) is isomorphic to a P-computable structure.