I came across the following questions regarding the time complexity and the computational cost. They are related to computations in non-abelian groups.
Suppose $H \rtimes_{\phi} K$ is a semidirect product of two finite groups, say $H$ and $K$. For an element $(h,k) \in H \rtimes_{\phi} K$, the inverse is $(\phi_k^{-1}(h^{-1}), k^{-1})$. If a person chooses any element $(h,k) \in H \rtimes_{\phi} K$ and compute its inverse, how can I explain about the time complexity and the computational cost of that computation?
And next if he computes, $(\phi_k^{-1}(h^{-1}), k^{-1})(h_1,k_1)$ for some $(h_1,k_1) \in H \rtimes_{\phi} K$, then what is the time complexity and the computational cost?
$(\phi_k^{-1}(h^{-1}), k^{-1})(h_1,k_1) = (\phi_k^{-1}(h^{-1}) \phi_{k^{-1}}(h_1), k^{-1} k_1)$
Thanks a lot in advance.