Definition:Irreducible (Representation Theory)/G-Module

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.

Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: 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". You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.