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abelian group additive group angle axes axis basis elements called classes coefficients commutative complex consider containing cosets cospace counterimage criterion curve defined diagram dimensional direct product direct sum direction cosines eigenvectors endomorphisms equation equivalence relation factor group fibre finite group Frobenius Algebra functions given group elements group G group multiplication hence hermitean Hurwitz integral idempotent identity intersection invariant irreducible representation isomorphic kernel Lecture left ideal linear combination mapping means minimal ideal monomials nilpotent ideal normal subgroup notation operator domain operator homomorphism order relation orthogonality relations outer product perpendicular Pierce decomposition polynomials possible projection regular representation represented right ideal ring elements rotation group satisfy scalar sided ideal similarity transformation stable subgroup sub ring submodule subring subset subspace summands Suppose theorem trace two-sided ideal unit matrix unitary matrix vector space verify write zero