The points which form the first row need not be written in their natural order, e.g.
Keeping this in mind, we call a permutation a cyclic permutation or a cycle if and only if it can be written in the form
where . In order to emphasize we also call it an r-cycle . We note that in this case the orbits of the subgroup generated by this permutation are the following subsets of : . We therefore abbreviate this cycle by , where the points which are cyclically permuted are put together in round brackets. For example . Commas which seperate the points may be omitted if no confusion can arise (e.g if ), and 1-cycles can be left out if it is clear which is meant. Hence we can write for the -cycle introduced above. This cycle can also be expressed in terms of alone: . Using all these abbreviations and denoting by the identity element, we obtain for example
There are the elements of the symmetric group in cycle notation. The notation for a cyclic permutation is not uniquely determined, since
2-cycles are called transpositions . The order of a cycle , i.e. the order of the cyclic group generated by this cycle, is equal to its length :
Two cycles and are called disjoint , if the two sets of points which are not fixed by and are disjoint sets. Notice that, for example, are disjoint cycles. Disjoint cycles and commute, i.e. . (We read compositions of mappings from right to left, so that .) Each permutation of a finite set can be written as a product of pairwise different disjoint cycles, e.g.
There are some more examples for the cycle decomposition of a permutation.
The disjoint cyclic factors of are uniquely determined by and therefore we call these factors together with the fixed point cycles of the cyclic factors of . Let denote the number of these cyclic factors of (including 1-cycles), let be their lengths, (recall that ), choose, for each an element of the -th cyclic factor. Then
This notation becomes uniquely determined if we choose the so that
If this holds, then is called the (standard) cycle notation for . We note in passing that the sets of points which are cyclically permuted by are just the orbits of the cyclic group generated by .