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

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The Dade group of a metacyclic $p$-group. / Mazza, Nadia.
In: Journal of Algebra, Vol. 266, No. 1, 01.08.2003, p. 102-111.

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Mazza N. The Dade group of a metacyclic $p$-group. Journal of Algebra. 2003 Aug 1;266(1):102-111. doi: 10.1016/S0021-8693(03)00328-4

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Mazza, Nadia. / The Dade group of a metacyclic $p$-group. In: Journal of Algebra. 2003 ; Vol. 266, No. 1. pp. 102-111.

Bibtex

@article{89917f0a9a154c8c807c7532715ad558,
title = "The Dade group of a metacyclic $p$-group.",
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.",
author = "Nadia Mazza",
note = "The final, definitive version of this article has been published in the Journal, Journal of Algebra 266 (1), 2003, {\textcopyright} ELSEVIER.",
year = "2003",
month = aug,
day = "1",
doi = "10.1016/S0021-8693(03)00328-4",
language = "English",
volume = "266",
pages = "102--111",
journal = "Journal of Algebra",
issn = "0021-8693",
publisher = "ELSEVIER ACADEMIC PRESS INC",
number = "1",

}

RIS

TY - JOUR

T1 - The Dade group of a metacyclic $p$-group.

AU - Mazza, Nadia

N1 - The final, definitive version of this article has been published in the Journal, Journal of Algebra 266 (1), 2003, © ELSEVIER.

PY - 2003/8/1

Y1 - 2003/8/1

N2 - 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.

AB - 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.

U2 - 10.1016/S0021-8693(03)00328-4

DO - 10.1016/S0021-8693(03)00328-4

M3 - Journal article

VL - 266

SP - 102

EP - 111

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -