Page 101 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Note that if f and g are disjoint, then fg=-gf, since f and g move disjoint sets of symbols.
Now let us examine one more example. Let h be the permutation in S defined by
5

 1 2 3 4 5
h =  4 2 3 5 1  .
 
Following our previous rules, we obtain
h = (1 4 5) (2) (3),
because each of the symbols 2 and 3 is left unchanged by h. By convention, we don’t include the
1-cycles (2) and (3) in the expression for h unless we wish to emphasize them, since they just
represent the identity permutation. Thus, we simply write h = (1 4 5).
The same process that we have just used is true for any cycle. That is, any r-cycle (i i . . . . . i ) can
r
1 2
be written as (i i ) (i i ) . . . . . (i i ), a product of transpositions.
2
1
1
1
r
1
Now we will use Theorem 2 to state a result which shows why transpositions are so important
in the theory of permutations.
Theorem 2: Every permutation in S (n  2) can be written as a product of transpositions.
n
Proof: The proof is really very simple. By Theorem 1 every permutation, apart from I, is a
product of disjoint cycles. Also, you have just seen that every cycle is a product of transpositions.
Hence, every permutation, apart from I, is a product of transpositions.
Also, I = (1 2) (1 2). Thus, I is also a product of transpositions. So, the theorem is proved.
Let us see how Theorem 3 works in practice. This is the same as (1 4) (1 2) (1 3) (1 5).

 1 2 3 4 5 6 
Similarly, the permutation  
 3 6 4 1 2 5 
= (1 3 4) (2 6 5) = (1 4) (1 3) (2 5) (2 6).
The decomposition given in Theorem 3 leads us to a subgroup of S that we will now discuss.
n
8.3 Alternating Group

You have seen that a permutation in S can be written as a product of transpositions. But all such
n
representations have one thing in common – if a permutation in S is the product of an odd
n
number of transpositions in one such representation, then it will be a product of an odd number
of transpositions in any such representation. Similarly, if f  S is a product of an even number
n
of transpositions in one representation, then f is a product of an even number of transpositions
in any such representation. To see this fact we need the concept of the signature or sign function.
Definition: The signature of f  S , (n  2) is defined to be
n
f(i) – f(i)
sign f  
i,j 1 j – i

For example, for f = (1 2 3)  S , 3

f(2) – f(1) f(3) – f(1) f(3) – f(2)
sign f = . .
2 – 1 3 – 1 3 – 2

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