# Groups of perfect self-isometries

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Let $B$ be a block of $\mathcal{O}G$ for a finite group $G$. The set of perfect isometries from $B$ to itself forms a group ${\rm PI}(B)$ under composition, and its isomorphism class is a derived invariant. Write ${\rm PI}_+(B)$ for the subgroup of ${\rm PI}(B)$ consisting of isometries with all signs positive (this is different to the subgroup ${\rm PI}^+(B)$ defined in [Ru11]). If ${\rm Piccent}(B)=1$, then ${\rm Pic}(B)$ embeds into ${\rm PI}_+(B)$.

The group ${\rm PI}(B)$ is useful in the calculation of Picard groups (see [EL18c]) and in the calculation of extensions of Morita equivalence classes from normal subgroups of index $p$ (see for example [Wa00] and [EL18a]).