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I’m studying the representations of finite groups.We all know that the group representations are very important tool in the study of finite groups by allowing many group theoretic problems to be reduced to problems in linear algebra. But, I don’t actually touch the link between “the group properties” and “the representation”! The reducibility of a representation, irreducibility of a representation, or completely reducibility of a representation…etc, are related to the vector space! So what about the group?! How can I understand the group more through the representation?!

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This isn't exactly what you are asking about, but I think you might find it interesting and helpful. What you are looking for sounds like this question:

But: In many ways the importance of groups is not the groups themselves, but how they act on vector spaces. That is, representation theory is an important topic in itself. One might even say that group theory is important because of its use in representation theory. One (standard) example of this is in the classification of elementary particles in physics. Here particles (made up of quarks) corresponds to representations (made up of irreducible representations) of certain groups. If you want to know more about this, then try to google "particle physics and representation theory". There is even a Wikipedia article about it.

Some other interesting and possibly helpful posts:

If you are interested in the general usefulness of representation theory, then you should take a look at Importance of Representation Theory.

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