Page 100 - DMTH403_ABSTRACT_ALGEBRA

Unit 8: Permutation Groups

In particular, 2-cycles are called transpositions. For example, the permutation f = (2 3)  S is a Notes
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transposition. Here f(1) = 1, f(2) = 3 and f(3) = 2.
Later you will see that transpositions play a very important role in the theory of permutations.

 1 2 .... n 
Now consider any 1-cycle (i) in S,. It is simply the identity permutation I    , since
 1 2 .... n 
it maps i to i and the other (n - 1) symbols to themselves.
Let us see some examples of cycles in S (1 2 3) is the 3-cycle that takes 1 to 2, 2 to 3 and 3 to 1. There
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are also 3 transpositions in S , namely, (1 2), (1 3) and (2 3).
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Now, can we express any permutation as a cycle? No. Consider the following example from S .
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Let g be the permutation defined by
 1 2 3 4 5 
g =  3 5 4 1 2  .
 
If we start with the symbol 1 and apply the procedure for obtaining a cycle to g, we obtain
(1 3 4) after three steps, Because, g maps 4 to 1, we close the brackets, even though we have not
yet written down all the symbols. Now we simply choose another symbol that has not appeared
so far, say 2, and start the procedure of writing a cycle again. Thus, we obtain another cycle (2 5).
Now, all the symbols are exhausted.

 g = (1 3 4) (2 5).
We call this expression for g a product of a 3-cycle and a transposition. In Figure 8.3 we represent
g by a diagram which shows the 3-cycle and the 2-cycle clearly.

Figure 8.3: (1 3 4) (2 5)

Because of the arbitrary choice of symbol at the beginning of each cycle, there are many ways of
expressing g. For example,
g = (4 1 3) (2 5) = (2 5) (1 3 4) = (5 2) (3 4 1).
That is, we can write the product of the separate cycles in any order, and the choice of the starting
element within each cycle is arbitrary.
So, you see that g can’t be written as a cycle; it is a product of disjoint cycles.
Definition: We call two cycle disjoint if they have no symbol in common. Thus, disjoint cycles
move disjoint sets of elements, (Note that f ! S,, moves a symbol i if f(i)  i. We say that f fixes
i if f(i) = i.)
So, for example, the cycles (1 2) and (3 4) in S are disjoint. But (1 2) and (1 4) are not disjoint, since
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they both move 1.

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