Let G be a group and A an abelian subgroup of G. I want to prove that for each irreducible representation $p$ we have that $dim(p)\leq [G:A]$.
I proved that for each subspace $p(A)-$invariant $W$, $W_0= \sum_{g \in G}p(g)(W)$ is a p(G)-invariant of V (hint of the professor) and that for a abelian finite group, each irreducible representation has dimension 1.
But how can i go on? I wish i can tell more of what i did in this question but i honestly have no idea how to proceed.I was thinking in the canonical decomposition of a representation, considering the subspaces W,but after some writing it seems non-sense.
Any help are welcome.
Ps:my representation theory course uses only knowlegde of theory of groups, not modules or other stuff.
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1$\begingroup$ Are you working in the “““easy””” case (finite groups, algebraically closed fields with chacteristics coprime with $|G|$), or are there subtleties? $\endgroup$– AphelliJul 24, 2020 at 14:59
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2$\begingroup$ The field needs to be algebraically closed, otherwise it's false. As $Z_3$ over $\mathbb{Q}$ demonstrates. As for your question, do you really need all $g\in G$ in the definition of $W_0$? In other words, what is $\dim(W_0)$? $\endgroup$– David A. CravenJul 24, 2020 at 15:07
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1$\begingroup$ I assume that you are not familiar with induced representations and Frobenius reciprocity? Because the result follows very neatly from that. $\endgroup$– Tobias KildetoftJul 25, 2020 at 11:12
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$\begingroup$ forgot to mention, the field is C and the group is finite, $\endgroup$– MyanklanaJul 25, 2020 at 13:53
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$\begingroup$ and no, i cant use induced representations for this, is literally the next content of the course $\endgroup$– MyanklanaJul 25, 2020 at 13:54
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You need some assumption on your group and the coefficient field.
I think the idea is to find a non-zero vector on which the abel group acts as a character. Then the remaining things work through: the representation is generated by the vector (as it is irreducible). As the abel subgroup acts on it as a character, we get a spanning set of the representation of order smaller than or equal to [G:H].
For example it is the case if your group is finite and the representation is taking coefficients in an algebraically closely field.
