Water's symmetry operations form the abelian group

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

Water's symmetry operations

E, C_2, \sigma_{xz}, \sigma_{yz}

form the abelian group

C_{2v}

, with the molecule in the

yz

plane and the

C_2

axis along

z

. Using only the group axioms, prove that every element of a group has exactly one inverse: if

h

and

h'

both satisfy

gh=hg=e

and

gh'=h'g=e

for the same element

g

, show that

h=h'

. ((a)) Give the general proof. ((b)) Take

g=\sigma_{yz}

, the reflection in the plane of the molecule. Write

C_2

\sigma_{xz}

and

\sigma_{yz}

3\times3

diagonal matrices, show that

\sigma_{yz}\sigma_{yz}=E

, and check directly that

E\sigma_{yz}

C_2\sigma_{yz}

and

\sigma_{xz}\sigma_{yz}

all differ from

E

, so that only

\sigma_{yz}

itself is its inverse.