MathDB

Subgroups with same size of invariant sets under all representations

Source: Alibaba Global Math Competition 2021, Problem 16

July 4, 2021

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Problem Statement

Let

G

be a finite group, and let

H_1, H_2 \subset G

be two subgroups. Suppose that for any representation of

G

on a finite-dimensional complex vector space

V

, one has that

\text{dim} V^{H_1}=\text{dim} V^{H_2},

where

V^{H_i}

is the subspace of

H_i

-invariant vectors in

V

(

i=1,2

). Prove that

Z(G) \cap H_1=Z(G) \cap H_2.

Here

Z(G)

denotes the center of

G