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In general, we know that if $G$ is a finite group and $K$ is a field, then $K[G]$ (the group algebra) is semisimple whenever $\operatorname{char}(K)$ does not divide the order of $G$. However, this result does not hold when we have a $p$-group and a finite field of characteristic $p$.

How would I go about showing that an irreducible representation of a $p$-group $G$ over a field $K$ of characteristic $p$ must be the trivial representation?

It seems like this would perhaps entail some sort of application of Maschke's Theorem to get a contradiction. I know that $|G| = \sum_i \operatorname{dim}(V_i)^2$, where $V_i$ is an irreducible representation. But it seems like that does not help me much here.

Any thoughts as to how to approach this question?

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Hint. Let $G$ act on the representation (minus $0$) and use an orbit counting argument to find a copy of the trivial representation.

Let $V$ be a nontrivial irreducible representation. Note that $V$ must be finite dimensional since $G$ is finite. Write $V^\circ = V\setminus\{0\}$ and let $G$ act on $V^\circ$. By orbit stabilizer, $\left| \mathcal{O}_x \right|$ divides $|G|$ for any $x\in V^\circ$, so all $G$-orbits have prime power (more precisely, power of $p$) size. Since these all need to sum to $\left|V^\circ\right|=p^n-1$, there is at least one orbit of size $1$. This orbit is a $G$-invariant one-dimensional subspace, and thus is an isomorphic copy of the trivial representation. But this contradicts that $V$ is irreducible.

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  • $\begingroup$ A very clean method of thinking of it. $\endgroup$ – Vladhagen Nov 19 '13 at 7:47
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    $\begingroup$ @Alexander Gruber I dont understand one thing. Why do you say that $|V^{\circ}|=p^n-1$? $\endgroup$ – Badshah Nov 1 '14 at 21:48
  • $\begingroup$ @alexander You're assuming $K = \mathbb{F}_p$, but he only said $K$ is of characteristic $p$. $\endgroup$ – Eric Auld Mar 31 '15 at 2:05
  • $\begingroup$ @Eric, see the comment by Mariano on this. $K$ contains $\mathbb{F}_p$ as the prime subfield, so we can define $$W=\mathrm{Span}_{\mathbb{F}_p}(\mathcal{O}(v))$$ where $\mathcal{O}(v)$ is the orbit of some non-zero $v\in V$. Such $W$, as a subset of $V$, is $G$-invariant, and so it is a representation of $G$ over $\mathbb{F}_p$, but it is not necessarily a subspace of $V$, and therefore might not be a sub-rep. I'm stuck here. Thoughts? $\endgroup$ – doodle Feb 19 '16 at 23:14
  • $\begingroup$ OK, I think I figured it out $\endgroup$ – doodle Feb 19 '16 at 23:52

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