In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups:

Lemma 3.16. Let $G$ be a compact group and $Rep(G)$ the category of finite dimensional unitary representations of $G$. For $\pi \in Rep(G)$ $H_\pi$ denotes the representation space of $\pi$. Suppose we have a Hilbert subspace $K_\pi\subset H_\pi$ for each $\pi\in Rep(G)$ satisfying the following: $$K_\pi\oplus K_\sigma \subset K_{\pi \oplus \sigma}, \quad \pi,\sigma \in Rep(G),$$ $$K_\pi\otimes K_\sigma \subset K_{\pi \otimes \sigma}, \quad \pi,\sigma \in Rep(G),$$ $$\overline{K_\pi}=K_{\overline{\pi}}, \quad \pi \in Rep(G),$$ where $\overline{\pi}$ is the complex conjugate representation and $\overline{K_\pi}$ is the image of $K_\pi$ under the natural map from $H_\pi$ to its complex conjugate Hilbert space. Then there exists a closed subgroup $H \subset G$ such that $$K_\pi=\{\xi \in H_\pi; \pi(h)\xi=\xi,\quad h\in H\}.$$

By applying this result to finite groups, we get a Galois correspondence between subgroups and subsystems (of unitary representations).

Such a result seems "fundamental" in the finite group theory but I don't know an "old" reference for it.

Question: What's the first reference for this result in finite group theory?


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