<s>
In	O
mathematics	O
,	O
a	O
permutation	B-Algorithm
group	I-Algorithm
is	O
a	O
group	O
G	O
whose	O
elements	O
are	O
permutations	B-Algorithm
of	O
a	O
given	O
set	O
M	O
and	O
whose	O
group	O
operation	O
is	O
the	O
composition	B-Application
of	O
permutations	B-Algorithm
in	O
G	O
(	O
which	O
are	O
thought	O
of	O
as	O
bijective	B-Algorithm
functions	I-Algorithm
from	O
the	O
set	O
M	O
to	O
itself	O
)	O
.	O
</s>
<s>
The	O
group	O
of	O
all	O
permutations	B-Algorithm
of	O
a	O
set	O
M	O
is	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
M	O
,	O
often	O
written	O
as	O
Sym(M )	O
.	O
</s>
<s>
The	O
term	O
permutation	B-Algorithm
group	I-Algorithm
thus	O
means	O
a	O
subgroup	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
If	O
then	O
Sym(M )	O
is	O
usually	O
denoted	O
by	O
Sn	O
,	O
and	O
may	O
be	O
called	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
n	O
letters	O
.	O
</s>
<s>
By	O
Cayley	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
,	O
every	O
group	O
is	O
isomorphic	O
to	O
some	O
permutation	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
way	O
in	O
which	O
the	O
elements	O
of	O
a	O
permutation	B-Algorithm
group	I-Algorithm
permute	O
the	O
elements	O
of	O
the	O
set	O
is	O
called	O
its	O
group	O
action	O
.	O
</s>
<s>
Being	O
a	O
subgroup	O
of	O
a	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
all	O
that	O
is	O
necessary	O
for	O
a	O
set	O
of	O
permutations	B-Algorithm
to	O
satisfy	O
the	O
group	O
axioms	O
and	O
be	O
a	O
permutation	B-Algorithm
group	I-Algorithm
is	O
that	O
it	O
contain	O
the	O
identity	O
permutation	B-Algorithm
,	O
the	O
inverse	O
permutation	B-Algorithm
of	O
each	O
permutation	B-Algorithm
it	O
contains	O
,	O
and	O
be	O
closed	O
under	O
composition	B-Application
of	O
its	O
permutations	B-Algorithm
.	O
</s>
<s>
A	O
general	O
property	O
of	O
finite	O
groups	O
implies	O
that	O
a	O
finite	O
nonempty	O
subset	O
of	O
a	O
symmetric	B-Algorithm
group	I-Algorithm
is	O
again	O
a	O
group	O
if	O
and	O
only	O
if	O
it	O
is	O
closed	O
under	O
the	O
group	O
operation	O
.	O
</s>
<s>
The	O
degree	O
of	O
a	O
group	O
of	O
permutations	B-Algorithm
of	O
a	O
finite	O
set	O
is	O
the	O
number	B-Application
of	I-Application
elements	I-Application
in	O
the	O
set	O
.	O
</s>
<s>
The	O
order	O
of	O
a	O
group	O
(	O
of	O
any	O
type	O
)	O
is	O
the	O
number	B-Application
of	I-Application
elements	I-Application
(	O
cardinality	B-Application
)	O
in	O
the	O
group	O
.	O
</s>
<s>
By	O
Lagrange	O
's	O
theorem	O
,	O
the	O
order	O
of	O
any	O
finite	O
permutation	B-Algorithm
group	I-Algorithm
of	O
degree	O
n	O
must	O
divide	O
n	O
!	O
</s>
<s>
since	O
n-factorial	O
is	O
the	O
order	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
Sn	O
.	O
</s>
<s>
Since	O
permutations	B-Algorithm
are	O
bijections	B-Algorithm
of	O
a	O
set	O
,	O
they	O
can	O
be	O
represented	O
by	O
Cauchy	O
's	O
two-line	O
notation	O
.	O
</s>
<s>
This	O
notation	O
lists	O
each	O
of	O
the	O
elements	O
of	O
M	O
in	O
the	O
first	O
row	O
,	O
and	O
for	O
each	O
element	O
,	O
its	O
image	O
under	O
the	O
permutation	B-Algorithm
below	O
it	O
in	O
the	O
second	O
row	O
.	O
</s>
<s>
If	O
is	O
a	O
permutation	B-Algorithm
of	O
the	O
set	O
then	O
,	O
</s>
<s>
Permutations	B-Algorithm
are	O
also	O
often	O
written	O
in	O
cycle	B-Algorithm
notation	I-Algorithm
(	O
cyclic	O
form	O
)	O
so	O
that	O
given	O
the	O
set	O
M	O
=	O
{	O
1	O
,	O
2	O
,	O
3	O
,	O
4}	O
,	O
a	O
permutation	B-Algorithm
g	O
of	O
M	O
with	O
g(1 )	O
=	O
2	O
,	O
g(2 )	O
=	O
4	O
,	O
g(4 )	O
=	O
1	O
and	O
g(3 )	O
=	O
3	O
will	O
be	O
written	O
as	O
(	O
1	O
,	O
2	O
,	O
4	O
)	O
(	O
3	O
)	O
,	O
or	O
more	O
commonly	O
,	O
(	O
1	O
,	O
2	O
,	O
4	O
)	O
since	O
3	O
is	O
left	O
unchanged	O
;	O
if	O
the	O
objects	O
are	O
denoted	O
by	O
single	O
letters	O
or	O
digits	O
,	O
commas	O
and	O
spaces	O
can	O
also	O
be	O
dispensed	O
with	O
,	O
and	O
we	O
have	O
a	O
notation	O
such	O
as	O
(	O
124	O
)	O
.	O
</s>
<s>
The	O
product	O
of	O
two	O
permutations	B-Algorithm
is	O
defined	O
as	O
their	O
composition	B-Application
as	O
functions	O
,	O
so	O
is	O
the	O
function	O
that	O
maps	O
any	O
element	O
x	O
of	O
the	O
set	O
to	O
.	O
</s>
<s>
Note	O
that	O
the	O
rightmost	O
permutation	B-Algorithm
is	O
applied	O
to	O
the	O
argument	O
first	O
,	O
because	O
of	O
the	O
way	O
function	B-Application
composition	I-Application
is	O
written	O
.	O
</s>
<s>
Some	O
authors	O
prefer	O
the	O
leftmost	O
factor	O
acting	O
first	O
,	O
but	O
to	O
that	O
end	O
permutations	B-Algorithm
must	O
be	O
written	O
to	O
the	O
right	O
of	O
their	O
argument	O
,	O
often	O
as	O
a	O
superscript	O
,	O
so	O
the	O
permutation	B-Algorithm
acting	O
on	O
the	O
element	O
results	O
in	O
the	O
image	O
.	O
</s>
<s>
However	O
,	O
this	O
gives	O
a	O
different	O
rule	O
for	O
multiplying	O
permutations	B-Algorithm
.	O
</s>
<s>
This	O
convention	O
is	O
commonly	O
used	O
in	O
the	O
permutation	B-Algorithm
group	I-Algorithm
literature	O
,	O
but	O
this	O
article	O
uses	O
the	O
convention	O
where	O
the	O
rightmost	O
permutation	B-Algorithm
is	O
applied	O
first	O
.	O
</s>
<s>
Since	O
the	O
composition	B-Application
of	O
two	O
bijections	B-Algorithm
always	O
gives	O
another	O
bijection	B-Algorithm
,	O
the	O
product	O
of	O
two	O
permutations	B-Algorithm
is	O
again	O
a	O
permutation	B-Algorithm
.	O
</s>
<s>
In	O
two-line	O
notation	O
,	O
the	O
product	O
of	O
two	O
permutations	B-Algorithm
is	O
obtained	O
by	O
rearranging	O
the	O
columns	O
of	O
the	O
second	O
(	O
leftmost	O
)	O
permutation	B-Algorithm
so	O
that	O
its	O
first	O
row	O
is	O
identical	O
with	O
the	O
second	O
row	O
of	O
the	O
first	O
(	O
rightmost	O
)	O
permutation	B-Algorithm
.	O
</s>
<s>
The	O
product	O
can	O
then	O
be	O
written	O
as	O
the	O
first	O
row	O
of	O
the	O
first	O
permutation	B-Algorithm
over	O
the	O
second	O
row	O
of	O
the	O
modified	O
second	O
permutation	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
given	O
the	O
permutations	B-Algorithm
,	O
</s>
<s>
The	O
composition	B-Application
of	O
permutations	B-Algorithm
,	O
when	O
they	O
are	O
written	O
in	O
cycle	B-Algorithm
notation	I-Algorithm
,	O
is	O
obtained	O
by	O
juxtaposing	O
the	O
two	O
permutations	B-Algorithm
(	O
with	O
the	O
second	O
one	O
written	O
on	O
the	O
left	O
)	O
and	O
then	O
simplifying	O
to	O
a	O
disjoint	O
cycle	O
form	O
if	O
desired	O
.	O
</s>
<s>
Since	O
function	B-Application
composition	I-Application
is	O
associative	O
,	O
so	O
is	O
the	O
product	O
operation	O
on	O
permutations	B-Algorithm
:	O
.	O
</s>
<s>
Therefore	O
,	O
products	O
of	O
two	O
or	O
more	O
permutations	B-Algorithm
are	O
usually	O
written	O
without	O
adding	O
parentheses	O
to	O
express	O
grouping	O
;	O
they	O
are	O
also	O
usually	O
written	O
without	O
a	O
dot	O
or	O
other	O
sign	O
to	O
indicate	O
multiplication	O
(	O
the	O
dots	O
of	O
the	O
previous	O
example	O
were	O
added	O
for	O
emphasis	O
,	O
so	O
would	O
simply	O
be	O
written	O
as	O
)	O
.	O
</s>
<s>
The	O
identity	O
permutation	B-Algorithm
,	O
which	O
maps	O
every	O
element	O
of	O
the	O
set	O
to	O
itself	O
,	O
is	O
the	O
neutral	O
element	O
for	O
this	O
product	O
.	O
</s>
<s>
In	O
cycle	B-Algorithm
notation	I-Algorithm
,	O
e	O
=	O
(	O
1	O
)	O
(	O
2	O
)	O
(	O
3	O
)	O
...(n )	O
which	O
by	O
convention	O
is	O
also	O
denoted	O
by	O
just	O
(	O
1	O
)	O
or	O
even	O
(	O
)	O
.	O
</s>
<s>
Since	O
bijections	B-Algorithm
have	O
inverses	O
,	O
so	O
do	O
permutations	B-Algorithm
,	O
and	O
the	O
inverse	O
σ−1	O
of	O
σ	O
is	O
again	O
a	O
permutation	B-Algorithm
.	O
</s>
<s>
Having	O
an	O
associative	O
product	O
,	O
an	O
identity	O
element	O
,	O
and	O
inverses	O
for	O
all	O
its	O
elements	O
,	O
makes	O
the	O
set	O
of	O
all	O
permutations	B-Algorithm
of	O
M	O
into	O
a	O
group	O
,	O
Sym(M )	O
;	O
a	O
permutation	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Consider	O
the	O
following	O
set	O
G1	O
of	O
permutations	B-Algorithm
of	O
the	O
set	O
M	O
=	O
{	O
1	O
,	O
2	O
,	O
3	O
,	O
4}	O
:	O
</s>
<s>
This	O
is	O
the	O
identity	O
,	O
the	O
trivial	O
permutation	B-Algorithm
which	O
fixes	O
each	O
element	O
.	O
</s>
<s>
This	O
permutation	B-Algorithm
interchanges	O
1	O
and	O
2	O
,	O
and	O
fixes	O
3	O
and	O
4	O
.	O
</s>
<s>
This	O
permutation	B-Algorithm
,	O
which	O
is	O
the	O
composition	B-Application
of	O
the	O
previous	O
two	O
,	O
exchanges	O
simultaneously	O
1	O
with	O
2	O
,	O
and	O
3	O
with	O
4	O
.	O
</s>
<s>
G1	O
forms	O
a	O
group	O
,	O
since	O
aa	O
=	O
bb	O
=	O
e	O
,	O
ba	O
=	O
ab	O
,	O
and	O
abab	O
=	O
e	O
.	O
This	O
permutation	B-Algorithm
group	I-Algorithm
is	O
,	O
as	O
an	O
abstract	O
group	O
,	O
the	O
Klein	O
group	O
V4	O
.	O
</s>
<s>
The	O
symmetries	O
are	O
determined	O
by	O
the	O
images	O
of	O
the	O
vertices	O
,	O
that	O
can	O
,	O
in	O
turn	O
,	O
be	O
described	O
by	O
permutations	B-Algorithm
.	O
</s>
<s>
The	O
rotation	O
by	O
90°	O
(	O
counterclockwise	O
)	O
about	O
the	O
center	O
of	O
the	O
square	O
is	O
described	O
by	O
the	O
permutation	B-Algorithm
(	O
1234	O
)	O
.	O
</s>
<s>
This	O
permutation	B-Algorithm
group	I-Algorithm
is	O
known	O
,	O
as	O
an	O
abstract	O
group	O
,	O
as	O
the	O
dihedral	B-Algorithm
group	I-Algorithm
of	O
order	O
8	O
.	O
</s>
<s>
In	O
the	O
above	O
example	O
of	O
the	O
symmetry	O
group	O
of	O
a	O
square	O
,	O
the	O
permutations	B-Algorithm
"	O
describe	O
"	O
the	O
movement	O
of	O
the	O
vertices	O
of	O
the	O
square	O
induced	O
by	O
the	O
group	O
of	O
symmetries	O
.	O
</s>
<s>
Any	O
such	O
homomorphism	O
is	O
called	O
a	O
(	O
permutation	B-Algorithm
)	O
representation	O
of	O
G	O
on	O
M	O
.	O
</s>
<s>
For	O
any	O
permutation	B-Algorithm
group	I-Algorithm
,	O
the	O
action	O
that	O
sends	O
(	O
g	O
,	O
x	O
)	O
→	O
g(x )	O
is	O
called	O
the	O
natural	O
action	O
of	O
G	O
on	O
M	O
.	O
This	O
is	O
the	O
action	O
that	O
is	O
assumed	O
unless	O
otherwise	O
indicated	O
.	O
</s>
<s>
The	O
action	O
of	O
a	O
group	O
G	O
on	O
a	O
set	O
M	O
is	O
said	O
to	O
be	O
transitive	O
if	O
,	O
for	O
every	O
two	O
elements	O
s	O
,	O
t	O
of	O
M	O
,	O
there	O
is	O
some	O
group	O
element	O
g	O
such	O
that	O
g(s )	O
=	O
t	O
.	O
Equivalently	O
,	O
the	O
set	O
M	O
forms	O
a	O
single	O
orbit	O
under	O
the	O
action	O
of	O
G	O
.	O
Of	O
the	O
examples	O
above	O
,	O
the	O
group	O
{	O
e	O
,	O
(	O
1	O
2	O
)	O
,	O
(	O
3	O
4	O
)	O
,	O
(	O
1	O
2	O
)	O
(	O
3	O
4	O
)	O
}	O
of	O
permutations	B-Algorithm
of	O
{	O
1	O
,	O
2	O
,	O
3	O
,	O
4}	O
is	O
not	O
transitive	O
(	O
no	O
group	O
element	O
takes	O
1	O
to	O
3	O
)	O
but	O
the	O
group	O
of	O
symmetries	O
of	O
a	O
square	O
is	O
transitive	O
on	O
the	O
vertices	O
.	O
</s>
<s>
A	O
permutation	B-Algorithm
group	I-Algorithm
G	O
acting	O
transitively	O
on	O
a	O
non-empty	O
finite	O
set	O
M	O
is	O
imprimitive	O
if	O
there	O
is	O
some	O
nontrivial	O
set	O
partition	O
of	O
M	O
that	O
is	O
preserved	O
by	O
the	O
action	O
of	O
G	O
,	O
where	O
"	O
nontrivial	O
"	O
means	O
that	O
the	O
partition	O
is	O
n't	O
the	O
partition	O
into	O
singleton	O
sets	O
nor	O
the	O
partition	O
with	O
only	O
one	O
part	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
full	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
a	O
set	O
M	O
is	O
always	O
primitive	O
.	O
</s>
<s>
That	O
is	O
,	O
f(g, x )	O
=	O
gx	O
for	O
all	O
g	O
and	O
x	O
in	O
G	O
.	O
For	O
each	O
fixed	O
g	O
,	O
the	O
function	O
fg(x )	O
=	O
gx	O
is	O
a	O
bijection	B-Algorithm
on	O
G	O
and	O
therefore	O
a	O
permutation	B-Algorithm
of	O
the	O
set	O
of	O
elements	O
of	O
G	O
.	O
Each	O
element	O
of	O
G	O
can	O
be	O
thought	O
of	O
as	O
a	O
permutation	B-Algorithm
in	O
this	O
way	O
and	O
so	O
G	O
is	O
isomorphic	O
to	O
a	O
permutation	B-Algorithm
group	I-Algorithm
;	O
this	O
is	O
the	O
content	O
of	O
Cayley	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
.	O
</s>
<s>
The	O
action	O
of	O
G1	O
on	O
itself	O
described	O
in	O
Cayley	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
gives	O
the	O
following	O
permutation	B-Algorithm
representation	O
:	O
</s>
<s>
The	O
bijection	B-Algorithm
λ	O
between	O
the	O
sets	O
is	O
given	O
by	O
.	O
</s>
<s>
The	O
study	O
of	O
groups	O
originally	O
grew	O
out	O
of	O
an	O
understanding	O
of	O
permutation	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
Permutations	B-Algorithm
had	O
themselves	O
been	O
intensively	O
studied	O
by	O
Lagrange	O
in	O
1770	O
in	O
his	O
work	O
on	O
the	O
algebraic	O
solutions	O
of	O
polynomial	O
equations	O
.	O
</s>
<s>
This	O
subject	O
flourished	O
and	O
by	O
the	O
mid	O
19th	O
century	O
a	O
well-developed	O
theory	O
of	O
permutation	B-Algorithm
groups	I-Algorithm
existed	O
,	O
codified	O
by	O
Camille	O
Jordan	O
in	O
his	O
book	O
Traité	O
des	O
Substitutions	O
et	O
des	O
Équations	O
Algébriques	O
of	O
1870	O
.	O
</s>
<s>
When	O
Cayley	O
introduced	O
the	O
concept	O
of	O
an	O
abstract	O
group	O
,	O
it	O
was	O
not	O
immediately	O
clear	O
whether	O
or	O
not	O
this	O
was	O
a	O
larger	O
collection	O
of	O
objects	O
than	O
the	O
known	O
permutation	B-Algorithm
groups	I-Algorithm
(	O
which	O
had	O
a	O
definition	O
different	O
from	O
the	O
modern	O
one	O
)	O
.	O
</s>
<s>
Cayley	O
went	O
on	O
to	O
prove	O
that	O
the	O
two	O
concepts	O
were	O
equivalent	O
in	O
Cayley	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
.	O
</s>
<s>
Another	O
classical	O
text	O
containing	O
several	O
chapters	O
on	O
permutation	B-Algorithm
groups	I-Algorithm
is	O
Burnside	O
's	O
Theory	O
of	O
Groups	O
of	O
Finite	O
Order	O
of	O
1911	O
.	O
</s>
<s>
The	O
first	O
half	O
of	O
the	O
twentieth	O
century	O
was	O
a	O
fallow	O
period	O
in	O
the	O
study	O
of	O
group	O
theory	O
in	O
general	O
,	O
but	O
interest	O
in	O
permutation	B-Algorithm
groups	I-Algorithm
was	O
revived	O
in	O
the	O
1950s	O
by	O
H	O
.	O
Wielandt	O
whose	O
German	O
lecture	O
notes	O
were	O
reprinted	O
as	O
Finite	O
Permutation	B-Algorithm
Groups	I-Algorithm
in	O
1964	O
.	O
</s>
