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For a fixed (let say finite-, or Lie-, to respect the historical motivations) group, why does the study of all its linear representations over a fixed field, leads to some knowledge about its structure that wouldn't be posible to reach without this tool ? What are the fundamental key points to understand here ?

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closed as off-topic by Steven Landsburg, YCor, Sean Lawton, Lucia, Jan-Christoph Schlage-Puchta Apr 22 at 17:49

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  • $\begingroup$ Possible duplicate of Why we need to study representations of matrix groups? $\endgroup$ – Steven Landsburg Apr 22 at 13:53
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    $\begingroup$ From Herstein's Carus Monograph on Noncommutative Rings, at the head of the section on applying representation theory to obtain results about groups: "It is difficult to see or explain why this machinery when turned loose on a group works so effectively, but work indeed it does." $\endgroup$ – Steven Landsburg Apr 22 at 13:57
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    $\begingroup$ The purported duplicate is somewhat focussed to groups that are given as automorphism groups. Even if in a sense technically speaking it's the same (since every group is automorphism group of some structure), in practice, and especially for such a subjective question, it's another point of view. However, I voted for migration to MathSE. $\endgroup$ – YCor Apr 22 at 13:59
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    $\begingroup$ @Steven Landsburg : Oh, maybe this is why I didn't manage to understand the deep reasons of why it works so well...I thought it was because of a lack of knowledge from my part.. $\endgroup$ – gerard Apr 22 at 14:14