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The Dade group of a metacyclic $p$-group.

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<mark>Journal publication date</mark>1/08/2003
<mark>Journal</mark>Journal of Algebra
Issue number1
Number of pages10
Pages (from-to)102-111
<mark>Original language</mark>English


The Dade group $D(P)$ of a finite $p$-group $P$, formed by equivalence classes of endo-permutation modules, is a finitely generated abelian group. Its torsion-free rank equals the number of conjugacy classes of non-cyclic subgroups of $P$ and it is conjectured that every non-trivial element of its torsion subgroup $D^t(P)$ has order $2$, (or also $4$, in case $p=2$). The group $D^t(P)$ is closely related to the injectivity of the restriction map $\Res:T(P)\rightarrow\prod_E T(E)$ where $E$ runs over elementary abelian subgroups of $P$ and $T(P)$ denotes the group of equivalence classes of endo-trivial modules, which is still unknown for (almost) extra-special groups ($p$ odd). As metacyclic $p$-groups have no (almost) extra-special section, we can verify the above conjecture in this case. Finally, we compute the whole Dade group of a metacyclic $p$-group.

Bibliographic note

The final, definitive version of this article has been published in the Journal, Journal of Algebra 266 (1), 2003, © ELSEVIER.