<s>
In	O
mathematics	O
,	O
and	O
in	O
particular	O
in	O
group	O
theory	O
,	O
a	O
cyclic	B-Algorithm
permutation	I-Algorithm
(	O
or	O
cycle	O
)	O
is	O
a	O
permutation	B-Algorithm
of	O
the	O
elements	O
of	O
some	O
set	O
X	O
which	O
maps	O
the	O
elements	O
of	O
some	O
subset	O
S	O
of	O
X	O
to	O
each	O
other	O
in	O
a	O
cyclic	O
fashion	O
,	O
while	O
fixing	O
(	O
that	O
is	O
,	O
mapping	O
to	O
themselves	O
)	O
all	O
other	O
elements	O
of	O
X	O
.	O
</s>
<s>
Cycles	B-Algorithm
are	O
often	O
denoted	O
by	O
the	O
list	O
of	O
their	O
elements	O
enclosed	O
with	O
parentheses	O
,	O
in	O
the	O
order	O
to	O
which	O
they	O
are	O
permuted	O
.	O
</s>
<s>
For	O
example	O
,	O
given	O
X	O
=	O
{	O
1	O
,	O
2	O
,	O
3	O
,	O
4}	O
,	O
the	O
permutation	B-Algorithm
(	O
1	O
,	O
3	O
,	O
2	O
,	O
4	O
)	O
that	O
sends	O
1	O
to	O
3	O
,	O
3	O
to	O
2	O
,	O
2	O
to	O
4	O
and	O
4	O
to	O
1	O
(	O
so	O
S	O
=	O
X	O
)	O
is	O
a	O
4-cycle	O
,	O
and	O
the	O
permutation	B-Algorithm
(	O
1	O
,	O
3	O
,	O
2	O
)	O
that	O
sends	O
1	O
to	O
3	O
,	O
3	O
to	O
2	O
,	O
2	O
to	O
1	O
and	O
4	O
to	O
4	O
(	O
so	O
S	O
=	O
{	O
1	O
,	O
2	O
,	O
3}	O
and	O
4	O
is	O
a	O
fixed	O
element	O
)	O
is	O
a	O
3-cycle	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
permutation	B-Algorithm
that	O
sends	O
1	O
to	O
3	O
,	O
3	O
to	O
1	O
,	O
2	O
to	O
4	O
and	O
4	O
to	O
2	O
is	O
not	O
a	O
cyclic	B-Algorithm
permutation	I-Algorithm
because	O
it	O
separately	O
permutes	O
the	O
pairs	O
{	O
1	O
,	O
3}	O
and	O
{	O
2	O
,	O
4}	O
.	O
</s>
<s>
Every	O
permutation	B-Algorithm
on	O
finitely	O
many	O
elements	O
can	O
be	O
decomposed	O
into	O
cycles	B-Algorithm
on	O
disjoint	B-Algorithm
orbits	O
.	O
</s>
<s>
The	O
individual	O
cyclic	O
parts	O
of	O
a	O
permutation	B-Algorithm
are	O
also	O
called	O
cycles	B-Algorithm
,	O
thus	O
the	O
second	O
example	O
is	O
composed	O
of	O
a	O
3-cycle	O
and	O
a	O
1-cycle	O
(	O
or	O
fixed	O
point	O
)	O
and	O
the	O
third	O
is	O
composed	O
of	O
two	O
2-cycles	O
,	O
and	O
denoted	O
(	O
1	O
,	O
3	O
)	O
(	O
2	O
,	O
4	O
)	O
.	O
</s>
<s>
thumb|upright	O
=	O
1.7	O
|Diagram	O
of	O
a	O
cyclic	B-Algorithm
permutation	I-Algorithm
with	O
two	O
fixed	O
points	O
;	O
a	O
6-cycle	O
and	O
two	O
1-cycles	O
.	O
</s>
<s>
A	O
permutation	B-Algorithm
is	O
called	O
a	O
cyclic	B-Algorithm
permutation	I-Algorithm
if	O
and	O
only	O
if	O
it	O
has	O
a	O
single	O
nontrivial	O
cycle	O
(	O
a	O
cycle	O
of	O
length	O
>	O
1	O
)	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
permutation	B-Algorithm
,	O
written	O
in	O
two-line	O
notation	O
(	O
in	O
two	O
ways	O
)	O
and	O
also	O
cycle	B-Algorithm
notation	I-Algorithm
,	O
</s>
<s>
Some	O
authors	O
restrict	O
the	O
definition	O
to	O
only	O
those	O
permutations	B-Algorithm
which	O
consist	O
of	O
one	O
nontrivial	O
cycle	O
(	O
that	O
is	O
,	O
no	O
fixed	O
points	O
allowed	O
)	O
.	O
</s>
<s>
is	O
a	O
cyclic	B-Algorithm
permutation	I-Algorithm
under	O
this	O
more	O
restrictive	O
definition	O
,	O
while	O
the	O
preceding	O
example	O
is	O
not	O
.	O
</s>
<s>
More	O
formally	O
,	O
a	O
permutation	B-Algorithm
of	O
a	O
set	O
X	O
,	O
viewed	O
as	O
a	O
bijective	B-Algorithm
function	I-Algorithm
,	O
is	O
called	O
a	O
cycle	O
if	O
the	O
action	O
on	O
X	O
of	O
the	O
subgroup	O
generated	O
by	O
has	O
at	O
most	O
one	O
orbit	O
with	O
more	O
than	O
a	O
single	O
element	O
.	O
</s>
<s>
A	O
cycle	O
can	O
be	O
written	O
using	O
the	O
compact	O
cycle	B-Algorithm
notation	I-Algorithm
(	O
there	O
are	O
no	O
commas	O
between	O
elements	O
in	O
this	O
notation	O
,	O
to	O
avoid	O
confusion	O
with	O
a	O
k-tuple	O
)	O
.	O
</s>
<s>
The	O
orbit	O
of	O
a	O
1-cycle	O
is	O
called	O
a	O
fixed	O
point	O
of	O
the	O
permutation	B-Algorithm
,	O
but	O
as	O
a	O
permutation	B-Algorithm
every	O
1-cycle	O
is	O
the	O
identity	O
permutation	B-Algorithm
.	O
</s>
<s>
When	O
cycle	B-Algorithm
notation	I-Algorithm
is	O
used	O
,	O
the	O
1-cycles	O
are	O
often	O
suppressed	O
when	O
no	O
confusion	O
will	O
result	O
.	O
</s>
<s>
One	O
of	O
the	O
basic	O
results	O
on	O
symmetric	B-Algorithm
groups	I-Algorithm
is	O
that	O
any	O
permutation	B-Algorithm
can	O
be	O
expressed	O
as	O
the	O
product	O
of	O
disjoint	B-Algorithm
cycles	B-Algorithm
(	O
more	O
precisely	O
:	O
cycles	B-Algorithm
with	O
disjoint	B-Algorithm
orbits	O
)	O
;	O
such	O
cycles	B-Algorithm
commute	O
with	O
each	O
other	O
,	O
and	O
the	O
expression	O
of	O
the	O
permutation	B-Algorithm
is	O
unique	O
up	O
to	O
the	O
order	O
of	O
the	O
cycles	B-Algorithm
.	O
</s>
<s>
The	O
multiset	B-Language
of	O
lengths	O
of	O
the	O
cycles	B-Algorithm
in	O
this	O
expression	O
(	O
the	O
cycle	O
type	O
)	O
is	O
therefore	O
uniquely	O
determined	O
by	O
the	O
permutation	B-Algorithm
,	O
and	O
both	O
the	O
signature	O
and	O
the	O
conjugacy	O
class	O
of	O
the	O
permutation	B-Algorithm
in	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
are	O
determined	O
by	O
it	O
.	O
</s>
<s>
The	O
number	O
of	O
k-cycles	O
in	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
Sn	O
is	O
given	O
,	O
for	O
,	O
by	O
the	O
following	O
equivalent	O
formulas	O
:	O
</s>
<s>
Since	O
disjoint	B-Algorithm
cycles	B-Algorithm
commute	O
,	O
the	O
inverse	O
of	O
a	O
product	O
of	O
disjoint	B-Algorithm
cycles	B-Algorithm
is	O
the	O
result	O
of	O
reversing	O
each	O
of	O
the	O
cycles	B-Algorithm
separately	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
permutation	B-Algorithm
that	O
swaps	O
2	O
and	O
4	O
.	O
</s>
<s>
Any	O
permutation	B-Algorithm
can	O
be	O
expressed	O
as	O
the	O
composition	B-Application
(	O
product	O
)	O
of	O
transpositions	O
—	O
formally	O
,	O
they	O
are	O
generators	O
for	O
the	O
group	O
.	O
</s>
<s>
In	O
fact	O
,	O
when	O
the	O
set	O
being	O
permuted	O
is	O
for	O
some	O
integer	O
,	O
then	O
any	O
permutation	B-Algorithm
can	O
be	O
expressed	O
as	O
a	O
product	O
of	O
and	O
so	O
on	O
.	O
</s>
<s>
The	O
decomposition	O
of	O
a	O
permutation	B-Algorithm
into	O
a	O
product	O
of	O
transpositions	O
is	O
obtained	O
for	O
example	O
by	O
writing	O
the	O
permutation	B-Algorithm
as	O
a	O
product	O
of	O
disjoint	B-Algorithm
cycles	B-Algorithm
,	O
and	O
then	O
splitting	O
iteratively	O
each	O
of	O
the	O
cycles	B-Algorithm
of	O
length	O
3	O
and	O
longer	O
into	O
a	O
product	O
of	O
a	O
transposition	O
and	O
a	O
cycle	O
of	O
length	O
one	O
less	O
:	O
</s>
<s>
This	O
means	O
the	O
initial	O
request	O
is	O
to	O
move	O
to	O
to	O
to	O
and	O
finally	O
to	O
Instead	O
one	O
may	O
roll	O
the	O
elements	O
keeping	O
where	O
it	O
is	O
by	O
executing	O
the	O
right	O
factor	O
first	O
(	O
as	O
usual	O
in	O
operator	O
notation	O
,	O
and	O
following	O
the	O
convention	O
in	O
the	O
article	O
Permutation	B-Algorithm
)	O
.	O
</s>
<s>
This	O
has	O
moved	O
to	O
the	O
position	O
of	O
so	O
after	O
the	O
first	O
permutation	B-Algorithm
,	O
the	O
elements	O
and	O
are	O
not	O
yet	O
at	O
their	O
final	O
positions	O
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
is	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
,	O
meaning	O
that	O
it	O
is	O
generated	O
by	O
elements	O
of	O
order	O
2	O
(	O
the	O
adjacent	O
transpositions	O
)	O
,	O
and	O
all	O
relations	O
are	O
of	O
a	O
certain	O
form	O
.	O
</s>
<s>
One	O
of	O
the	O
main	O
results	O
on	O
symmetric	B-Algorithm
groups	I-Algorithm
states	O
that	O
either	O
all	O
of	O
the	O
decompositions	O
of	O
a	O
given	O
permutation	B-Algorithm
into	O
transpositions	O
have	O
an	O
even	O
number	O
of	O
transpositions	O
,	O
or	O
they	O
all	O
have	O
an	O
odd	O
number	O
of	O
transpositions	O
.	O
</s>
<s>
This	O
permits	O
the	O
parity	O
of	O
a	O
permutation	B-Algorithm
to	O
be	O
a	O
well-defined	O
concept	O
.	O
</s>
