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Can prime fields act two ways on the same abelian group?

  1. May 24, 2012 #1
    A problem asks to find an abelian group [itex]V[/itex] and a field [itex]F[/itex] such that there exist two different actions, call them [itex]\cdot[/itex] and [itex]\odot[/itex], of [itex]F[/itex] on [itex]V[/itex] such that [itex]V[/itex] is an [itex]F[/itex]-module.

    A usual way to solve this is to take any vector space over a field with a non-trivial automorphism group, and define [itex]r\odot \mu[/itex] to be [itex]f(r)\cdot \mu[/itex] for [itex]f\in \text{Aut}(F), f\neq \iota[/itex].

    My question is: is this essentially the only way? Given two different actions of [itex]F[/itex] on [itex]V[/itex], can we construct a non-trivial automorphism of [itex]F[/itex]?

    Or, a weaker result: can we show that a prime field cannot act in two different ways on a module?
  2. jcsd
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