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

Mazza, Nadia (2003) The Dade group of a metacyclic $p$-group. Journal of Algebra, 266 (1). pp. 102-111. ISSN 0021-8693

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    Abstract

    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.

    Item Type: Article
    Journal or Publication Title: Journal of Algebra
    Additional Information: The final, definitive version of this article has been published in the Journal, Journal of Algebra 266 (1), 2003, © ELSEVIER.
    Subjects: Q Science > QA Mathematics
    Departments: Faculty of Science and Technology > Mathematics and Statistics
    ID Code: 20815
    Deposited By: Dr Nadia Mazza
    Deposited On: 03 Dec 2008 11:19
    Refereed?: No
    Published?: Published
    Last Modified: 09 Oct 2013 12:44
    Identification Number:
    URI: http://eprints.lancs.ac.uk/id/eprint/20815

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