A rigid equilateral triangular plate lies in the -plane

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 rigid equilateral triangular plate lies in the

xy

-plane, centred at the origin, with one vertex on the

+x

-axis. Its symmetry operations are

E

, rotations

C_3

(

120^\circ

) and

C_3^2

(

240^\circ

) about the

z

-axis, and three reflections in vertical planes each containing the

z

-axis and one vertex; call the one containing the

+x

-axis vertex

\sigma_1

. These six operations form a group isomorphic to

C_{3v}

, composed right to left (

gh

means

h

first). ((a)) Using only the group axioms, derive the general rule

(gh)^{-1}=h^{-1}g^{-1}

for any two elements

g,h

of a group. ((b)) Write

C_3

and

\sigma_1

3\times3

matrices, and verify the rule explicitly by computing

(\sigma_1C_3)^{-1}

two ways: directly, by showing

\sigma_1C_3

is its own inverse, and via the rule, as

C_3^{-1}\sigma_1^{-1}=C_3^2\sigma_1