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Action of a cyclic group on modules

  1. Jan 22, 2009 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations

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

    3. The attempt at a solution

    I noticed that no (nontrivial) element of [tex]M[/tex] is invariant under the action of [tex]\mathbb{Q}G[/tex]. 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
  2. jcsd
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