Difference between revisions of "Cycle (permutation)"
(added some) |
m |
||
(One intermediate revision by one other user not shown) | |||
Line 1: | Line 1: | ||
A '''cycle''' is a type of [[permutation]]. | A '''cycle''' is a type of [[permutation]]. | ||
− | Let <math>S_M</math> be the symmetric group on a [[set]] <math>M</math>. Let <math>\zeta</math> be an element of <math>S_M</math>, and let <math>\bar{\zeta}</math> be the [[subgroup]] of <math>S_M</math> generated by <math>\zeta</math>. Then <math>\zeta</math> is a '''cycle''' if <math>M</math> has exactly one [[orbit]] (under the operation of <math>\bar{\zeta}</math>) which does not consist of a single [[element]]. This orbit is called the ''support'' of <math>\zeta</math>, and is sometimes denoted <math>\text{supp}(\zeta | + | Let <math>S_M</math> be the symmetric group on a [[set]] <math>M</math>. Let <math>\zeta</math> be an element of <math>S_M</math>, and let <math>\bar{\zeta}</math> be the [[subgroup]] of <math>S_M</math> generated by <math>\zeta</math>. Then <math>\zeta</math> is a '''cycle''' if <math>M</math> has exactly one [[orbit]] (under the operation of <math>\bar{\zeta}</math>) which does not consist of a single [[element]]. This orbit is called the ''support'' of <math>\zeta</math>, and is sometimes denoted <math>\text{supp}(\zeta)</math>. |
== Some properties of cycles == | == Some properties of cycles == |
Latest revision as of 22:13, 12 January 2017
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.