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We show that the $p$-power maps in the first Hochschild cohomology space offinite-dimensional selfinjective algebras over a field of prime characteristic$p$ commute with stable equivalences of Morita type on the subgroup of classesrepresented by integrable derivations. We show, by giving an example, that the$p$-power maps do not necessarily commute with arbitrary transfer maps in theHochschild cohomology of symmetric algebras.

Invariance of the restricted $p$-power map on integrable derivations under stable equivalences