A square-based pyramid with base corners at and apex on the axis has the group of…

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

A square-based pyramid with base corners at

(\pm1,\pm1,0)

and apex on the

+z

axis has the group

C_{4v}

of order 8: the rotations

E

C_4

C_2=C_4^2

C_4^3

about

z

(

C_4

through

90^\circ

), the reflections

\sigma_v

\sigma_v'

in the planes

x=y

and

x=-y

, and the reflections

\sigma_d

\sigma_d'

in the planes

y=0

and

x=0

. Products read right to left. ((a)) Show that

\det

is a homomorphism from

C_{4v}

onto

\{+1,-1\}

. Find its kernel

N

and the order of

N

, and identify the quotient

C_{4v}/N

. ((b)) Someone proposes

C_{4v}=N\times\{E,\sigma_d\}

. Say which of the three conditions for a direct product hold and which fail. ((c)) Show that

C_{4v}

is not the direct product of any two of its subgroups, neither of which is

\{E\}

or the whole group.