# Can the fix point set of a nontrivial irreducible complex representation of a finite odd order group be non trivial?

I'm trying to show that if $$G$$ is a finite odd order group then, all of its nontrivial complex representations are of complex type (i.e., it is not realisable over the reals).

Let $$V$$ be such a representation, and let $$V^G:=\{v\in V: gv=v, \forall g \in G\}$$ be its fix point set. Using some theorems in Bröcker's book about Representations of Compact Lie Groups, I can solve this problem if I show that $$V^G=0$$. Is that true?

• Well, what about the trivial representation? – Matt Samuel Jan 14 '19 at 0:11
• It should be a non-trivial representation*, I'll change @MattSamuel – Andre Gomes Jan 14 '19 at 0:12
• I have answered the first question here: math.stackexchange.com/questions/1033844/… – Andre Gomes Jan 14 '19 at 17:24

Any nonzero vector $$v\in V^G$$ spans a $$1$$-dimensional trivial sub-representation. If $$V$$ is irreducible of dimension $$>1$$, then it follows that $$V^G=0$$. If $$\dim V=1$$, then $$V^G\neq 0$$ implies $$V$$ is the trivial representation.