A regular octagonal stop sign has symmetry group of order 16: eight rotations by…
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry
Invariance, Ch Groups: the algebra of symmetry operations
Problem
A regular octagonal stop sign has symmetry group C8v of order 16: eight rotations by multiples of 45∘ about the centre and eight reflections. Someone reasons: 'Writing σ1 for the reflection through one vertex and C8 for the 45∘ rotation, I compute σ1C8 as Mat(C8)Mat(σ1), because σ1 is written first, so it acts first. This gives a reflection matrix, so {E,σ1C8} is a subgroup of order 2. Since 8 divides 16, there must also be a subgroup of order 8 made entirely of reflections.' Find every error in this reasoning.