A square-based pyramid has its base corners at (in some length unit) and its apex on…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Group theory and symmetry

Invariance, Ch Groups: the algebra of symmetry operations

Problem

A square-based pyramid has its base corners at

(\pm1,\pm1,0)

(in some length unit) and its apex on the

+z

-axis, which is the principal axis. Its eight symmetry operations form the group

C_{4v}

: the identity

E

; rotations

C_4

C_2\,(=C_4^2)

C_4^3

about the

z

-axis; and four reflections in vertical planes:

\sigma_v

in the plane

x=y

and

\sigma_v'

in the plane

x=-y

(each containing two opposite corners), and

\sigma_d

in the plane

y=0

and

\sigma_d'

in the plane

x=0

(each containing the midpoints of two opposite base edges). Operations compose right to left (

gh

means

h

first). ((a)) Write

C_4

(rotation by

90^\circ

about

z

) and

\sigma_d

(reflection in the plane

y=0

) as

3\times3

matrices, and compute the products

\sigma_dC_4

and

C_4\sigma_d

. Identify each product with one of the eight named operations. ((b)) By Lagrange's theorem, list every order a subgroup of

C_{4v}

could have. ((c)) Show that the four rotations

H=\{E,C_4,C_2,C_4^3\}

form a subgroup of

C_{4v}

, and find its left cosets in

C_{4v}

: state how many there are and which of the eight operations each one contains.