Corollary 4

a) Suppose that the group G is finite and that K is a commutative field of characteristic 0. Two finite-dimensional linear representations of G are isomorphic if and only if they have the same character.

b) Suppose that K is a perfect field of characteristic a prime number $p$ and that the group G is finite, of cardinal prime to $p$. The character of a finite-dimensional linear representation of G is zero if and only if this representation is isomorphic to the direct sum of $p$ pairwise isomorphic representations.

Under the assumptions of the corollary, every finite-dimensional linear representation of G is semisimple. The corollary then follows from the corollary of VIII, p. 384.