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By applying Pontryagin's duality, $V^*$ becomes a discrete $G$-module. The appropriate reformulation of question 2 to this case is treated in Decomposing representations of finite groupsDecomposing representations of finite groups

An example of an action is given there so the answer to the question is "no", for $\mathbb{F}_p$ at least.

By applying Pontryagin's duality, $V^*$ becomes a discrete $G$-module. The appropriate reformulation of question 2 to this case is treated in Decomposing representations of finite groups

An example of an action is given there so the answer to the question is "no", for $\mathbb{F}_p$ at least.

By applying Pontryagin's duality, $V^*$ becomes a discrete $G$-module. The appropriate reformulation of question 2 to this case is treated in Decomposing representations of finite groups

An example of an action is given there so the answer to the question is "no", for $\mathbb{F}_p$ at least.

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Pablo
Pablo
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By applying Pontryagin's duality, $V^*$ becomes a discrete $G$-module. The appropriate reformulation of question 2 to this case is treated in Decomposing representations of finite groups

An example of an action is given there so the answer to the question is "no", for $\mathbb{F}_p$ at least.