Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 19I / c / iii

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- Solution iii

Solution

No. The center of a group satisfies

Z (C_{2} \times D_{8}) = C_{2} \times Z (D_{8}) ≅ C_{2} \times C_{2},

because

Z (D_{8}) = {1, r^{2}} ≅ C_{2}

. This centre is not cyclic. The cyclic-center obstruction to a faithful irreducible representation therefore rules out a faithful irreducible complex representation of

C_{2} \times D_{8}

.

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