Cycle (permutation)
A cycle is a type of permutation.
Let be the symmetric group on a set
. Let
be an element of
, and let
be the subgroup of
generated by
. Then
is a cycle if
has exactly one orbit (under the operation of
) which does not consist of a single element. This orbit is called the support of
, and is sometimes denoted
.
Some properties of cycles
Lemma. Let be a family of cycles of
with pairwise disjoint supports
. Then the
commute. The product
is then well defined as
, for
, and
, for
. Let
be the subgroup generated by
. Then the function
is a bijection from
to the orbits of
containing more than one element.
Proof. Suppose and
are of the
. Then
so by symmetry
. This proves that the
commute and justifies the definition of
.
Suppose is a an orbit of
with more than one element, and suppose
. Then by our characterization of
,
must belong to
, for some
; since
is the orbit of
, it follows that
. Thus the mapping
is a surjection from
to the orbits of
with more than one element; since it is evidently injective, it follows that this mapping is a bijection.
Theorem (cycle notation). Let be an element of
. Then there exists a unique set
of cycles of
with pairwise disjoint supports such that
Proof. Let be the subgroup of
generated by
. Let
be the family of nonempty orbits of
. For all
, let
be the restriction of
to
; let
. Then by the lemma,
Since the mapping
must be a bijection from
to the orbits of
, it follows from the lemma that
is unique.