In [this paper][1] (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? 


  [1]: http://www.sciencedirect.com/science/article/pii/S0022123697932286#