Applications of representation theories to the study of the represented objects

In the representation theory of finite groups we have Burnside's theorem about groups of order $p^aq^b$. The statement has nothing to do with representations, but the proof uses character theory to prove a result about finite groups.

Are there such results in representation theories of other objects? For example, is there a result about Lie groups which does not mention representations in its statement, by is proved through the representation theory of Lie groups? What about Lie algebra representation theory? What about representations of associative algebras?

Note: I am currently studying the basics of Lie groups, Lie algebras and their representations.

The classification of semisimple Lie algebras crucially relies on at least two pieces of representation theory, namely 1) a detailed study of the adjoint representation of a semisimple Lie algebra $\mathfrak{g}$ (this is where roots and the root system come from), and 2) knowledge of the classification of representations of $\mathfrak{sl}_2$. This in turn leads to a classification of e.g. compact Lie groups, after some more effort.