Ammonia's symmetry operations compose to form the group (products read right to left

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Groups: the algebra of symmetry operations

Problem

Ammonia's symmetry operations

E, C_3, C_3^2, \sigma_v, \sigma_v', \sigma_v''

compose to form the group

C_{3v}

(products read right to left, so

gh

means

h

first). Using only the four group axioms (closure, associativity, an identity, and inverses), prove that a group can contain only one identity element: if

e

and

e'

both satisfy

eg = ge = g

and

e'g = ge' = g

for every element

g

of the group, show that

e = e'

. State, in one sentence, which of ammonia's six operations this identifies, and why there cannot physically be two different 'leave everything where it is' operations.

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