# Homework Help: Action of a cyclic group on modules

1. Jan 22, 2009

### Storm22

1. The problem statement, all variables and given/known data

Let $$G = <x>$$ be a cyclic group of prime order $$p$$ and let $$M$$ be a vector space over $$\mathbb{Q}$$ with basis $$S = \{m_0,m_1,\dots,m_{p-1}\}$$. $$G$$ acts on the $$S$$ in a natural way by cyclic permutations and this action is linearly extended to an action of $$G$$ on $$M$$. Now, the resulting action is extended (linearly) to an action of $$\mathbb{Q}G$$ (the group ring) on $$M$$. Denote $$v_i = m_i - m_{i-1}$$ for $$i=1,2,\dots,p-1$$ and denote $$M' = \mathbb{Q}v_1 + \dots + \mathbb{Q}v_{p-1}$$ (module sum). Prove that $$M'$$ cannot be written as the direct sum of two nontrivial $$\mathbb{Q}G$$-modules.

2. Relevant equations

$$\mathbb{Q}G$$ is defined to be the set of formal sums of elements of $$G$$, with coefficients from $$\mathbb{Q}$$.

3. The attempt at a solution

I noticed that no (nontrivial) element of $$M$$ is invariant under the action of $$\mathbb{Q}G$$. Thought of showing that an invariant element must always exist in case the result is false, but haven't managed to do that. I tried showing weaker/stronger statements (that M' is cyclic or simple), but had no luck with that either.

Last edited: Jan 22, 2009