# Simple module with non-simple restriction.

If $$G$$ is a group and $$H$$ a subgroup of $$G$$, would it be possible to have a simple $$\mathbb{C}G$$-module $$V$$ for which the restriction $$\operatorname{Res}^G_HV$$ is not a simple $$\mathbb{C}H$$-module? What would be examples of such a behaviour?

• More or less every triple $(G,H,V)$ satisfies this. For example, any simple module of dimension greater than $1$ and any abelian subgroup. – David A. Craven Jul 31 at 21:03
• Okay, lets say $V$ is an irreducible $\mathbb{C}G$-module with $dim(V)=2$ for example. Now lets say $H$ is an abelian subgroup of $G$. We can consider $H$ as a $\mathbb{C}H$-module. Since $dim(V)>1$ we can decompose it in two vector spaces $V=V_1\oplus V_2$. But why are they submodules? If $v_1\in V_1$, do we necessarily have $h\cdot v_1\in V_1$? – roi_saumon Jul 31 at 21:34
• All simple modules for abelian groups are $1$-dimensional. Not all subspaces are submodules. But they are there somewhere. – David A. Craven Jul 31 at 21:39
• Oh, I get it now. Thanks – roi_saumon Jul 31 at 21:55

If we take for $$H$$ the trivial subgroup of $$G$$ then the restriction $$\operatorname{Res}^G_H V$$ is simple if and only if $$V$$ is one-dimensional. In this way every simple $$\mathbb{C}G$$-module of dimension at least two gives a counterexample.