Someone studying the symmetry group of a square-based pyramid (order 8: the identity…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Groups: the algebra of symmetry operations

Problem

Someone studying the

C_{4v}

symmetry group of a square-based pyramid (order 8: the identity

E

; rotations

C_4

C_2 = C_4^2

C_4^3

about the vertical axis by

90^\circ

180^\circ

270^\circ

; and four reflections,

\sigma_v

\sigma_v'

through opposite corners of the base and

\sigma_d

\sigma_d'

through opposite edge midpoints of the base) argues: 'A conjugacy class is a single fixed rotation considered on its own. Turning the pyramid by

90^\circ

is a different operation from turning it by

270^\circ

, so

C_4

and

C_4^3

must lie in different conjugacy classes.' ((a)) Taking

C_4

as rotation by

90^\circ

about

z

and

\sigma_v

as reflection in the plane containing the

x

-axis and

z

-axis, compute the matrix for

\sigma_v C_4 \sigma_v^{-1}

and use it to decide whether the argument is correct. ((b)) List all the conjugacy classes of

C_{4v}

with their sizes.