Authors: Serge Bouc and Nadia Romero.

Title: The Whitehead group of (almost) extra-special $p$-groups with $p$ odd.

Abstract: Let $p$ be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special $p$-groups. For this we compute, for any finite $p$-group~$P$, the subgroup $Cl_1(\Z P)$ of $SK_1(\Z P)$, in terms of a genetic basis of $P$. We also introduce a deflation map from $Cl_1(\Z P)$ to $Cl_1(\Z(P/N))$, for a normal subgroup $N$ of $P$, and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of $SK_1(\Z P)$, when $P$ is an elementary abelian $p$-group.