= Solution
Choose a nonzero vector in an <irreducible representation>. Its <finite group> orbit spans a finite-dimensional <invariant subspace>, which must be the whole representation. The representing matrices of an <abelian group> commute. Choose an <eigenspace> of one matrix; all other matrices preserve it. Repeatedly choosing an <eigenspace> for the restrictions of the remaining matrices gives a common <eigenvector>. Its line is invariant under the whole group. Irreducibility therefore makes the representation that line, proving \b[every irreducible complex representation of a finite <abelian group> is one-dimensional].
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