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Next: Order of a permutation Up: Groups Previous: SUBGROUPS

PERMUTATION GROUPS

An important class of groups are permutation groups. One reason for their importance is that every group may be represented as a group of permutations on a suitable set.

Let A be a set, then a permutation of A is a bijection tex2html_wrap_inline903 . Read the notes on functions if you are unfamiliar with this idea.

If A is finite then we may as well let tex2html_wrap_inline907 and we write such a permutation as

displaymath909

where the tex2html_wrap_inline911 are distinct elements of A.

For example, let tex2html_wrap_inline915 . There are six permutations for this set, namely tex2html_wrap_inline917 , tex2html_wrap_inline919 , tex2html_wrap_inline921 , tex2html_wrap_inline923 , tex2html_wrap_inline925 , tex2html_wrap_inline927 .

There are two other common ways to write such a permutation. If the set A is understood to be a set of consecutive integers (or an ordered set) then the top line is deleted so we would write

displaymath931

where it is understood that tex2html_wrap_inline933 maps 1 to tex2html_wrap_inline935 etc. We shall not use this way of writing permutations. The second way, and the method we shall use in these notes, is to write the permutation as a product of disjoint cycles. A cycle is constructed as follows: Choose some starting element, say tex2html_wrap_inline937 . Now compute the elements tex2html_wrap_inline939 , tex2html_wrap_inline941 , and so on until we arrive back at the element i (this is guaranteed if A is finite). Enclose this list of elements of A in parentheses to form the cycle tex2html_wrap_inline949 where by tex2html_wrap_inline951 we mean tex2html_wrap_inline933 applied to i k-times. Now repeat the process to form the next cycle choosing as starting element an element of A that has not appeared in any previous cycle. The process ends when every element of A appears in exactly one cycle. The representation of tex2html_wrap_inline933 is then obtained by juxtaposing (multiplying) these disjoint cycles. It is usual to suppress cycles containing only one element.

As an example, if tex2html_wrap_inline965 here are two of the three ways we discussed to write a certain permutation

displaymath967

where in the cycle notation we have suppressed the cycles (4)(7)(8). In a cycle such as tex2html_wrap_inline969 we mean that the permutation maps 1 to 2, maps 2 to 3 and maps 3 to 1.

Note that the cycle tex2html_wrap_inline969 can also be written as tex2html_wrap_inline973 since it contains the same information, but could not have been written as tex2html_wrap_inline975 .

The six permutations on tex2html_wrap_inline915 written as permutations in cycle form are 1, tex2html_wrap_inline981 , tex2html_wrap_inline983 , tex2html_wrap_inline985 , tex2html_wrap_inline969 , tex2html_wrap_inline989 .




next up previous
Next: Order of a permutation Up: Groups Previous: SUBGROUPS

Peter Williams
Sun Mar 30 14:48:35 PST 1997