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 .
