3.4 Subgroups
29
Table 3.4 Multiplication table for the symmetric group S 3 . The unit element can also be expressed
as three 1-cycles: (A)(B)(C)
S 3
ˆ
E
(ABC)
(ACB)
(A)(BC)
(B)(AC)
(C)(AB)
ˆ
E
ˆ
E
(ABC)
(ACB)
(A)(BC)
(B)(AC)
(C)(AB)
(ABC)
(ABC)
(ACB)
ˆ
E
(C)(AB)
(A)(BC)
(B)(AC)
(ACB)
(ACB)
ˆ
E
(ABC)
(B)(AC)
(C)(AB)
(A)(BC)
(A)(BC) (A)(BC)
(B)(AC)
(C)(AB)
ˆ
E
(ABC)
(ACB)
(B)(AC) (B)(AC)
(C)(AB)
(A)(BC)
(ACB)
ˆ
E
(ABC)
(C)(AB) (C)(BA)
(A)(BC)
(B)(AC)
(ABC)
(ACB)
ˆ
E
The group multiplication table contains all there is to know about a group. It
hides a wealth of internal structure that is directly relevant to the physical phenomena to which the group applies. In order to elucidate this structure, three ways of
delineating subsets of the group are useful: subgroups, cosets, and classes.
3.4 Subgroups
A subgroup H of G, denoted H ⊂ G, is a subset of elements of G, which itself has
the group property. Trivial subgroups are the group containing the identity alone,
denoted as C 1 ={ ˆ
E}, and the group G itself. Besides these, in the case of the group
of ammonia, C 3v , there are four nontrivial subgroups: C 3 ={ ˆ
E, ˆ
C 3 , ˆ
C 2
3 }, and C s =
{ ˆ
E, ˆ
σ i } with i = 1, 2, or 3. The three C s groups are equivalent. We can construct
a simplified genealogical tree, which shows the subgroup structure (Fig. 3.4). In
chemistry and physics, subgroup structures are highly relevant since the distortions
of a symmetric system can be described as a descent down the genealogical tree. We
shall describe this in Sect. 4.6 as the subduction process. For the moment, we retain
Cayley’s theorem:
Theorem 1 Every group of order n is isomorphic with a subgroup of the symmetric
group S n .
This theorem is immediately clear from the multiplication table. A given row of
the table shows how the corresponding element maps the entire set of elements onto
itself. This mapping is a permutation of the n elements, and every element gives
rise to a different permutation since no two rows are the same. Thus, G must be
a subgroup of S n . The importance of this theorem is especially evident from the
mathematical point of view. It tells us that the symmetric groups exhaust all the
possible structures of finite groups.
29
Table 3.4 Multiplication table for the symmetric group S 3 . The unit element can also be expressed
as three 1-cycles: (A)(B)(C)
S 3
ˆ
E
(ABC)
(ACB)
(A)(BC)
(B)(AC)
(C)(AB)
ˆ
E
ˆ
E
(ABC)
(ACB)
(A)(BC)
(B)(AC)
(C)(AB)
(ABC)
(ABC)
(ACB)
ˆ
E
(C)(AB)
(A)(BC)
(B)(AC)
(ACB)
(ACB)
ˆ
E
(ABC)
(B)(AC)
(C)(AB)
(A)(BC)
(A)(BC) (A)(BC)
(B)(AC)
(C)(AB)
ˆ
E
(ABC)
(ACB)
(B)(AC) (B)(AC)
(C)(AB)
(A)(BC)
(ACB)
ˆ
E
(ABC)
(C)(AB) (C)(BA)
(A)(BC)
(B)(AC)
(ABC)
(ACB)
ˆ
E
The group multiplication table contains all there is to know about a group. It
hides a wealth of internal structure that is directly relevant to the physical phenomena to which the group applies. In order to elucidate this structure, three ways of
delineating subsets of the group are useful: subgroups, cosets, and classes.
3.4 Subgroups
A subgroup H of G, denoted H ⊂ G, is a subset of elements of G, which itself has
the group property. Trivial subgroups are the group containing the identity alone,
denoted as C 1 ={ ˆ
E}, and the group G itself. Besides these, in the case of the group
of ammonia, C 3v , there are four nontrivial subgroups: C 3 ={ ˆ
E, ˆ
C 3 , ˆ
C 2
3 }, and C s =
{ ˆ
E, ˆ
σ i } with i = 1, 2, or 3. The three C s groups are equivalent. We can construct
a simplified genealogical tree, which shows the subgroup structure (Fig. 3.4). In
chemistry and physics, subgroup structures are highly relevant since the distortions
of a symmetric system can be described as a descent down the genealogical tree. We
shall describe this in Sect. 4.6 as the subduction process. For the moment, we retain
Cayley’s theorem:
Theorem 1 Every group of order n is isomorphic with a subgroup of the symmetric
group S n .
This theorem is immediately clear from the multiplication table. A given row of
the table shows how the corresponding element maps the entire set of elements onto
itself. This mapping is a permutation of the n elements, and every element gives
rise to a different permutation since no two rows are the same. Thus, G must be
a subgroup of S n . The importance of this theorem is especially evident from the
mathematical point of view. It tells us that the symmetric groups exhaust all the
possible structures of finite groups.