×

Roots of simple modules. (English) Zbl 1102.20006

Let \(F\) be an algebraically closed field of characteristic \(p>0\) and \(G\) a finite group. Let \(U\) be an indecomposable \(FG\)-module with vertex \(D\) and source \(Z\). Then one can associate to these data a projective indecomposable \(F[DC_G(D)/D]\)-module \(\widetilde R\), called a root of \(U\). This module is unique up to \(N_G(D,Z)/D\)-conjugation. Observe also that \(U\) is relatively \(DC_G(D)\)-projective, so there is an indecomposable \(F[DC_G(D)]\)-module \(Y\) with vertex \(D\) such that \(Y\mid\text{Res}^G_{DC_G(D)}U\) and \(U\mid\text{Ind}^G_{DC_G(D)}Y\). Let \(J=N_G(D,Y)\) be the inertial group of \(Y\) in \(N_G(D)\).
The author investigates the properties of roots of indecomposables and then proves that if \(U\) is simple, then \(O_p(J/DC_G(D))=1\). He also points out that in P. Landrock’s book [Finite group algebras and their modules. Lond. Math. Soc. Lect. Note Ser. 84. Cambridge: Cambridge Univ. Press (1983; Zbl 0523.20001)] it is incorrectly claimed that \(J/DC_G(D)\) is a \(p'\)-group. As a corollary, it is shown that the following result of K. Erdmann [Bull. Lond. Math. Soc. 9, 216-218 (1977; Zbl 0389.20010)] is still true. If \(U\) is simple with cyclic vertex \(D\) and belongs to the block \(B\) of \(FG\), then \(D\) is also a defect group of \(B\).

MSC:

20C20 Modular representations and characters
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
PDFBibTeX XMLCite
Full Text: DOI