Page 99 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Remark: From now we will refer to the composition of permutations as multiplication of
permutations. We will also drop the composition sign. Thus, we will write f o g as fg.
The two-line notation that we have used for a permutation is rather cumbersome. In the next
section we will see how to use a shorter notation.

8.2 Cyclic Decomposition

Let us first discuss what a cycle is.

 2 4 
Consider the permutation f =  1 2  . Choose any one of the symbols say 1.
 
Now, we write down a left hand bracket followed by I : (1
Since f maps 1 to 3, we write 3 after 1 : (1 3
Since f maps 3 to 4, we write 4 after 3 : (1 3 4

Since f maps 4 to 2, we write 2 after 4 : ( l 3 4 2
Since f maps 2 to 1 (the symbol we started with),
we close the brackets after the symbol (1 3 4 2)

Thus, we write f = (1 3 4 2). This means that f maps each symbol to the symbol on its right, except
for the final symbol in the brackets, which is mapped to the first.
If we had chosen 3 as our starting symbol we would have obtained the expression (3 4 2 1) for f.
However, this means exactly the same as (1 3 4 2), because both denote the permutation which
we have represented diagrammatically in Figure 8.2.
Figure 8.2: (1 3 4 2)

1 3

2 4

Such a permutation is called a 4-cycle, or a cycle of length 4. Figure 8.2 can give you an indication
as to why we give this name.
Let us give a definition now.

Definition: A permutation f  S , is called an r-cycle (or cycle of length r) if there are r distinct
n
integers i , i ,, i , . . . , i lying between 1 and n such that
3
r
2
1
f(i ) = i , f(i ) = i , . . . . . , f( ) = i , f(i ) = i . 1
r
i,-1
r
1
2
2
3
and f(k) = k  k  {i , i , . . . , i ).
r
2
1
Then, we write f = (i i . . . . . i ).
r
1 2
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