Mathematical Methods for Physicists: A concise introduction - Site Map

SUBGROUPS AND COSETS
is replaced by 2. So by the combined action 1 is replaced by 2 and we have the ®rst
column
1
:
2
We leave the other two columns to be completed by the reader.
Each element of a group has an inverse. Thus, for each permutation P i there is
P 1
i , the inverse of P i . We can use the property P i P 1
i ˆ P 1 to ®nd P 1
i . Let us ®nd
P 1
6 :
P 1
6 ˆ 3 1 2
1 2 3
ˆ 1 2 3
ˆ P 2 :
2 3 1
It is straightforward to check that
P 6 P 1
6 ˆ P 6 P 2 ˆ 1 2 3
1 2 3
3 1 2 2 3 1
ˆ 1 2 3
ˆ P 1 :
1 2 3
The reader can verify that our group S 3 is generated by the elements P 2 and P 3 ,
while P 1 serves as the identity. This means that the other three distinct elements
can be expressed as distinct multiplicative combinations of P 2 and P 3 :
P 4 ˆ P 2 2P 3 ; P 5 ˆ P 2 P 3 ; P 6 ˆ P 2 2:
The symmetric group S n plays an important role in the study of ®nite groups.
Every ®nite group of order n is isomorphic to a subgroup of the permutation
group S n . This is known as Cayley's theorem. For a proof of this theorem the
interested reader is referred to an advanced text on group theory.
In physics, these permutation groups are of considerable importance in the
quantum mechanics of identical particles, where, if we interchange any two or
more these particles, the resulting con®guration is indistinguishable from the
original one. Various quantities must be invariant under interchange or permutation
of the particles. Details of the consequences of this invariant property may be
found in most ®rst-year graduate textbooks on quantum mechanics that cover the
application of group theory to quantum mechanics.
Subgroups and cosets
A subset of a group G, which is itself a group, is called a subgroup of G. This idea
was introduced earlier. And we also saw that C 3 , a cyclic group of order 3, is a
subgroup of S 3 , a symmetric group of order 6. We note that the order of C 3 is a
factor of the order of S 3 . In fact, we will show that, in general,
the order of a subgroup is a factor of the order of the full group
(that is, the group from which the subgroup is derived).
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