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Publication# Torsion endo-trivial modules

Abstract

We prove that the group T(G) of endo-trivial modules for a non-cyclic finite p-group G is detected on restriction to the family of subgroups which are either elementary abelian of rank 2 or (almost) extraspecial. This result is closely related to the problem of finding the torsion subgroup of T(G). We give the complete structure of T(G) when G is dihedral, semi-dihedral, or quaternion.

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Related concepts (7)

Finite group

In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just

Torsion subgroup

In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order (the torsion elements of A). An abelian group A i

Elementary abelian group

In mathematics, specifically in group theory, an elementary abelian group is an abelian group in which all elements other than the identity have the same order. This common order must be a prime num

Related publications (1)

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The group D(P) of all endo-permutation modules for a finite p-group P is a finitely generated abelian group. We prove that its torsion-free rank is equal to the number of conjugacy classes of non-cyclic subgroups of P. We also obtain partial results on its torsion subgroup. We determine next the structure of Q\otimes D(-) viewed as a functor, which turns out to be a simple functor S_{E,Q}, indexed by the elementary group E of order p^2 and the trivial Out(E)-module Q. Finally we describe a rather strange exact sequence relating Q\otimes D(P), Q\otimes B(P), and Q\otimes R(P), where B(P) is the Burnside ring and R(P) is the Grothendieck ring of QP-modules.

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