Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 10 / c

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- Solution c

Solution

Act on vectors in the plane of the regular hexagon using its actual rotations and reflections. Choose the first reflection axis as the horizontal axis. A faithful representation is generated by

D_{2} (R) = (1/2 3 /2 - 3 /2 1/2), D_{2} (m_{1}) = (10 0 - 1) .

They are real orthogonal matrices obeying

D_{2} (R)^{6} = I

,

D_{2} (m_{1})^{2} = I

and

D_{2} (m_{1}) D_{2} (R) D_{2} (m_{1}) = D_{2} (R)^{- 1}

. Hence they reproduce the group relations. Since

m_{2} = R m_{1}

,

D_{2} (m_{2}) = (1/2 3 /2 3 /2 - 1/2) .

This is reflection in the line at angle

π /6

. More generally

D_{2} (m_{j}) = D_{2} (R)^{j - 1} D_{2} (m_{1})

. The six rotation matrices are distinct, as are the six reflection matrices; their determinant signs distinguish the two sets. Thus no two group elements have the same matrix, proving faithfulness rather than merely the representation property.

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