Endo-permutation modules as sources of simple modules.

The source of a simple $kG$-module, for a finite $p$-solvable group $G$ and an algebraically closed field $k$ of prime characteristic $p$, is an endo-permutation module (see~\cite{Pu1} or~\cite{Th}). L. Puig has proved, more precisely, that this source must be isomorphic to the cap of an endo-permutation module of the form \bigotimes_{Q/R\in\cal S}\Ten^P_Q\Inf^Q_{Q/R}(M_{Q/R}), where $M_{Q/R}$ is an indecomposable torsion endo-trivial module with vertex $Q/R$, and $\cal S$ is a set of cyclic, quaternion and semi-dihedral sections of the vertex of the simple $kG$-module. At present, it is conjectured that, if the source of a simple module is an endo-permutation module, then it should have this shape. In this paper, we are going to give a method that allow us to realize explicitly the cap of any such indecomposable module as the source of a simple module for a finite $p$-nilpotent group

The Dade group of a metacyclic pp-group.

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

Endotrivial modules for the sporadic simple groups and their covers

In a step towards the classification of endotrivial modules for quasi-simple groups, we investigate endotrivial modules for the sporadic simple groups and their covers. A main outcome of our study is the existence of torsion endotrivial modules with dimension greater than one for several sporadic groups with $p$-rank greater than one.Comment: in Journal of Pure and Applied Algebra, published online 19th February 201

Endotrivial modules for the Schur covers of the symmetric and alternating groups

We investigate the endotrivial modules for the Schur covers of the symmetric and alternating groups and determine the structure of their group of endotrivial modules in all characteristics. We provide a full description of this group by generators and relations in all cases.Comment: changes from (v21): corrected minor typos, added reference

The group of endotrivial modules for the symmetric and alternating groups.

We complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that, for n ≥ p2, the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. The torsion subgroup is trivial for the alternating groups. The torsion-free part of the group is free abelian of rank 1 if n ≥ p2 + p and has rank 2 if p2 ≤ n < p2 + p. This completes the work begun earlier by Carlson, Mazza and Nakano

Endotrivial modules for the symmetric and alternating groups.

In this paper we determine the group of endotrivial modules for certain symmetric and alternating groups in characteristic $p$. If $p=2$, then the group is generated by the class of $\Omega^n(k)$ except in a few low degrees. If $p >2$, then the group is only determined for degrees less than $p^2$. In these cases we show that there are several Young modules which are endotrivial

Endotrivial Modules for the General Linear Group in a Nondefining Characteristic

Suppose that $G$ is a finite group such that $\operatorname{SL}(n,q)\subseteq G \subseteq \operatorname{GL}(n,q)$, and that $Z$ is a central subgroup of $G$. Let $T(G/Z)$ be the abelian group of equivalence classes of endotrivial $k(G/Z)$-modules, where $k$ is an algebraically closed field of characteristic~$p$ not dividing $q$. We show that the torsion free rank of $T(G/Z)$ is at most one, and we determine $T(G/Z)$ in the case that the Sylow $p$-subgroup of $G$ is abelian and nontrivial. The proofs for the torsion subgroup of $T(G/Z)$ use the theory of Young modules for $\operatorname{GL}(n,q)$ and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules

Endotrivial modules for finite groups of Lie type A in nondefining characteristic

Let $G$ be a finite group such that \SL(n,q)\subseteq G \subseteq \GL(n,q) and $Z$ be a central subgroup of $G$. In this paper we determine the group $T(G/Z)$ consisting of the equivalence classes of endotrivial $k(G/Z)$-modules where $k$ is an algebraically closed field of characteristic $p$ such that $p$ does not divide $q$. The results in this paper complete the classification of endotrivial modules for all finite groups of (untwisted) Lie Type $A$, initiated earlier by the authors