Definition:Irreducible (Representation Theory)/G-Module
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Although this article appears correct, it's inelegant. There has to be a better way of doing it.
Write the above in more rigorous language: either "they are equivalent" (in the sense that there exists a bijection between them) or "there exists a bijection between them".
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Definition
A $G$-module is irreducible if and only if the corresponding linear representation is irreducible.
That is, any proper $G$-submodule is trivial.
Also see
In Correspondence between Linear Group Actions and Linear Representations, it is shown that linear representations and $G$-modules are bijective.
Write the above in more rigorous language: either "they are equivalent" (in the sense that there exists a bijection between them) or "there exists a bijection between them".
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