On $ \rho $-conjugate Hopf-Galois structures

The Hopf-Galois structures admitted by a Galois extension of fields $L/K$ with Galois group $G$ correspond bijectively with certain subgroups of $\mathrm{Perm}(G)$. We use a natural partition of the set of such subgroups to obtain a method for partitioning the set of corresponding Hopf-Galois structures, which we term $\rho$-conjugation. We study properties of this construction, with particular emphasis on the Hopf-Galois analogue of the Galois correspondence, the connection with skew left braces, and applications to questions of integral module structure in extensions of local or global fields. In particular, we show that the number of distinct $\rho$-conjugates of a given Hopf-Galois structure is determined by the corresponding skew left brace, and that if $H, H'$ are Hopf algebras giving $\rho$-conjugate Hopf-Galois structures on a Galois extension of local or global fields $L/K$ then an ambiguous ideal $\mathfrak{B}$ of $L$ is free over its associated order in $H$ if and only if it is free over its associated order in $H'$. We exhibit a variety of examples arising from interactions with existing constructions in the literature.