The molecular symmetry operations must be closed under composition. Write the proposed nonidentity operations as

R = C_{2}

and

S = σ

. Their product

RS

cannot equal any proposed member:

RS = E \Rightarrow R = S^{- 1} = S, RS = R \Rightarrow S = E, RS = S \Rightarrow R = E .

(1)

Each conclusion contradicts the distinct operations stipulated. Therefore at least a fourth symmetry operation must exist. For a mirror plane containing the twofold axis, the product generates another mirror plane; for a plane perpendicular to it, the product is inversion. A tilted plane may generate still more operations, but cannot evade group closure.

ii

 0  0

Solution

 0  0

One choice is the C2v molecular point group,

{E, C_{2} (z), σ_{x z}, σ_{yz}}

. The additional operation is a second reflection, since

C_{2} (z) σ_{x z} = σ_{yz} .

(1)

A bent water molecule has this symmetry. Choose the

z

axis along its bond-angle bisector and put the molecule in the

x z

plane: reflection in

yz

exchanges its two hydrogen atoms, while reflection in

x z

fixes their positions.

iii

 0  0

Solution

 0  0

With

R = C_{2} (z)

S = σ_{x z}

and

T = σ_{yz}

, composition gives

E R S T E E R S T R R E T S S S T E R T T S R E

(1)

Every entry belongs to the set, proving closure.

E

is the identity element; every element is its own inverse. Associativity follows because these are actual spatial transformations: composing functions is associative. Thus all group axioms hold, and the group is isomorphic to the Klein four-group.

For a concrete matrix check, use