MathDB

16

Part of 2021 Alibaba Global Math Competition

Problems(1)

Subgroups with same size of invariant sets under all representations

Source: Alibaba Global Math Competition 2021, Problem 16

7/4/2021
Let GG be a finite group, and let H1,H2⊂GH_1, H_2 \subset G be two subgroups. Suppose that for any representation of GG on a finite-dimensional complex vector space VV, one has that dimVH1=dimVH2,\text{dim} V^{H_1}=\text{dim} V^{H_2}, where VHiV^{H_i} is the subspace of HiH_i-invariant vectors in VV (i=1,2i=1,2). Prove that Z(G)∩H1=Z(G)∩H2.Z(G) \cap H_1=Z(G) \cap H_2. Here Z(G)Z(G) denotes the center of GG.
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