In this work we deal with representations of finite groups. First, we say that a representation is a homomorphism from a group G to a linear group GL(V) and the degree of the representation is equal to the dimension of the space V. We then tell what G-invariant subspaces and irreducible representations are. We further define the equivalence relation between the representations and prove that the irreducibility and decomposability are preserved on the equivalence classes. The first major goal is to prove that any representation of a finite group is equivalent to the direct sum of irreducible representations. Then, with the theory of representations, we prove some results from linear algebra. We prove that any irreducible representation of the abelian group is of the first degree. Finally, we look at the characters of the representations. The character is a mapping that returns the trace of the representation evaluated at element for each element in the group. We show that we can tell with characters whether representation is irreducible. The last major result, however, is that any representation is equivalent to exactly one direct sum of irreducible representations, with the individual elements in the sum determined to the equivalence class exactly.
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