\$\mathbb{Z}/p^k \mathbb{Z}[G]\$-modules in repr. theory - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:03:49Z http://mathoverflow.net/feeds/question/110910 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110910/mathbbz-pk-mathbbzg-modules-in-repr-theory \$\mathbb{Z}/p^k \mathbb{Z}[G]\$-modules in repr. theory Mike Beyman 2012-10-28T15:39:00Z 2013-06-10T04:22:00Z <p>My question is referring to the answer in <a href="http://mathoverflow.net/questions/106543/p-adic-representations-of-groups" rel="nofollow">this link</a></p> <p>especially to this sentence:</p> <p>"if \$G\$ is a finite group and \$M\$ is a \$Z_p[G]\$-module, then \$M\$ is (in)decomposable if and only if \$M/p^kM\$ is a (in)decomposable \$Z/p^kZ[G]\$-module for some \$k\$."</p> <p>Why is it easier to determine the number of the indecomposable \$Z/p^kZ[G]\$-modules for some \$k\$ which have the form \$M/p^kM\$. Do I really have to search first for the indecomposable \$M/p^kM\$-modules for each \$k=1,\cdots,d\$, with \$d\$ minimal in \$1=p^d\$. And then look if one can write this modules in the form \$M/p^kM\$? This seams quite difficult for me.</p> <p>Thank you for hints how to solve this.</p> http://mathoverflow.net/questions/110910/mathbbz-pk-mathbbzg-modules-in-repr-theory/110919#110919 Answer by Simone Virili for \$\mathbb{Z}/p^k \mathbb{Z}[G]\$-modules in repr. theory Simone Virili 2012-10-28T16:55:09Z 2012-10-28T16:55:09Z <p>well... one can start thinking to the case when \$G\$ is trivial. You can probably feel how \$\mathbb Z/p^k\mathbb Z\$-modules are easier then \$\mathbb Z_p\$-modules. In particular, the torsion \$\mathbb Z_p\$-modules are the abelian \$p\$-groups, while the \$\mathbb Z/p^k\mathbb Z\$-modules are the \$p^k\$-bounded abelian \$p\$-groups. It seems more than reasonable to start classifying the indecomposable bounded groups in order to classify the torsion ones... when \$G\$ is not trivial, the situation is similar.</p>