Question

If there is a finite group $G$ with a cyclic normal subgroup $C_n$, one can describe the indecomposable representations of $G$ through induction. How does $Ind_{C_n}^G$ decompose? For representations over fields, I know that Clifford's theory whould help. But what happens if the representations should be over a ring? I am interested expecially in the p-adic ring $Z_p$.

Best regards

Answer 1

Not really an answer, but this is already difficult in the complex case, when $C$ is a central subgroup. For example, if $C = Z(G),$ and we induce a faithful irreducible $C$-module to $G,$ the number of distinct irreducible constituents is bounded above by the number of conjugacy classes of $G/C,$ but I don't know many other general statements- the number can be as low as $1$ (when we have a character of so-called ``central type"). To understand the general case of inducing an indecomposable $RC$-module to $G$ for a general commutative ring $R$ when $C$ ne not be central, Mackey decompoosition is certainly a helpful tool ( the induced module restricts freely to any subgroup meeting $C$ only in the identity, and more generally, the restriction to a subgroup $H$ of the induced module can be reasonably described by knowing how the original indecomposable restricts to $C \cap H,$ though precise details may require care. When working over $\mathbb{Z}_{p}$, Green's indecomposability theorem may also be useful.