Groups - Group Theory Applied to Chemistry - page 30

representatives as powers of the threefold axis, which allows introduction of a cyclic
summation:

3

( C 3 ) n C s C 3 v = (3.21)

n = 1

This expression forms a nice illustration of how representatives make different maps
of the subgroup. These cosets form an orbit inside the group. Note that the way
cosets are defined here is based on left multiplication by the generators, giving rise
to what are also denoted as left cosets . An analogous partitioning of the group can
also be based on right cosets. The partitioning of a group in cosets gives rise to the
famous Lagrange theorem:

Theorem 2 The order of a subgroup of a finite group is a divisor of the order of the
group .

Consider a group G and subgroup H with respective orders | G | and | H | .The

theorem states that is an integer. The proof is based on two elements. One

first has to prove that all elements in a given coset are different and further that
different cosets do not manifest any overlap.
Consider a coset R i H with elements R i h x .For h x = h y , R i h x must be different
from R i h y , simply because two elements in the same row in the multiplication table
can never be equal, as was proven in Eq. ( 3.9 ). Hence, the size of the coset will be
equal to

| G | / | H |

∈ R i H . This new element will in turn
be the representative of a new coset, R j H , and we must prove that this new coset
does not overlap with the previous one. This can easily be demonstrated by reductio
ad absurdum . Suppose that there is an element R j h x in the second coset that also
belongs to the first coset, as R i h y . We then have:

. Then we consider an element R j /

| |

H

R j h x = R i h y
R − 1
i

(3.22)

R j = h y h − 1

x

Since the subgroup H has the group property, the inverse element h − 1
x

is also an

element of H , and so is its product with h y . Hence, the product R − 1

R j will be an

i

element of H ,say h z = h y h − 1 . The first coset will of course contain the element

x

R i h z , which reduces to

R i h z = R i h y h − x = R i R − 1 R j = R j

(3.23)

i

Hence, R j ∈ R i H , contrary to the assumption. The expansion of the group in cosets
thus leads to a complete partitioning in subsets of equal sizes. It starts by the subset
formed by the subgroup H .If H is smaller than G , take an element outside H and
form with this element a coset, which will have the same dimension as H . If there
are still elements outside, use one of these to form a new coset, again containing | H |