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Let $p$ be a prime number, and let $G$ be a finite group whose order is a power of $p$. Let $F$ be a field of characteristic $p$, and $V$ a nonzero vector space over $F$ equipped with a linear action of $G$. Does there exist a nonzero subspace $W \subset V$ such that G acts trivially on W?

I am stuck at this question. Now, a vector space on a finite field is again a finite field (I think so). Now, the finite field $F$ is itself a vector space over $F$. I think the group $G$ acts trivially on $F$. Is this right? Thanks beforehand.

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We want to show that $G$ has a fixed point on $V$. In order to do so, we let $0 \neq v \in V$ and consider the $\mathbb{F}_p$-span of the orbit of $v$ under $G$, let's call this space $V'$. This is a finite dimensional vector space over $\mathbb{F}_p$, thus $|V'| = p^n$ for some $n$. The group $G$ acts on $V'$ and we have a partition $$V' = \bigsqcup O_i $$ into $G$-orbits. Note that either $|O_i| = 1$, which means that the single element in $O_i$ is a fixed point or $p$ divides $|O_i|$ (orbit-stabilizer-theorem). The identity $$ \sum_{i} |O_i| \equiv 0 \text{ (mod }p)$$ yields that $\{i \:|\: |O_i| = 1\}$ is either empty or contains at least $p$ elements. However, since $0 \in V'$ is a fixed point, it follows that this set is not empty and thus there exists at least one non-zero fixed point in $V'$. Take $W$ to be the subspace generated by this element and you are done.

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  • $\begingroup$ As much as I have understood, any field of characteristic $p$ contains $\mathbb{F}_P$, so you have considered a simpler case to produce a vector $v$ which is fixed by $G$. The action of $G$ is assumed to be a on vector space over $\mathbb{F}_p$. But linearity of the action ensures that the subspace generated by $v$ over $F$ is fixed. $\endgroup$
    – nkh99
    Mar 11, 2023 at 11:12
  • $\begingroup$ @nkh99 Yes, that is right. Essentially, two things have to be realized: One, the problem can be solved for vector spaces over $\mathbb{F}_p$ and two, the problem can be reduced to the case of considering vector spaces over $\mathbb{F}_p$. $\endgroup$ Mar 12, 2023 at 9:36

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