 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-2/10/a/solution

The table describes the symmetry group of a regular hexagon. Choose

R

as a counterclockwise rotation through

π /3

and

m_{1}

as reflection in one symmetry axis. With

s = m_{1}

, the relations are

R^{6} = I, s^{2} = I, s R s = R^{- 1} .

The rotation entries already are

I, R, \dots, R^{5}

, and the table gives

R m_{j} = m_{j + 1}

with indices modulo six. Therefore

m_{j} = R^{j - 1} s (j = 1, \dots, 6), G = {R^{k}, R^{k} s : 0 \leq k < 6} .

The source writes

D_{6}

with the subscript counting polygon vertices; this dihedral group has order 12.

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