<s>
In	O
group	O
theory	O
,	O
Cayley	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
,	O
named	O
in	O
honour	O
of	O
Arthur	O
Cayley	O
,	O
states	O
that	O
every	O
group	O
is	O
isomorphic	O
to	O
a	O
subgroup	O
of	O
a	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
More	O
specifically	O
,	O
is	O
isomorphic	O
to	O
a	O
subgroup	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
whose	O
elements	O
are	O
the	O
permutations	B-Algorithm
of	O
the	O
underlying	O
set	O
of	O
.	O
</s>
<s>
The	O
proof	O
of	O
Cayley	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
in	O
this	O
case	O
shows	O
that	O
if	O
is	O
a	O
finite	O
group	O
of	O
order	O
,	O
then	O
is	O
isomorphic	O
to	O
a	O
subgroup	O
of	O
the	O
standard	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
But	O
might	O
also	O
be	O
isomorphic	O
to	O
a	O
subgroup	O
of	O
a	O
smaller	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
for	O
some	O
;	O
for	O
instance	O
,	O
the	O
order	O
6	O
group	O
is	O
not	O
only	O
isomorphic	O
to	O
a	O
subgroup	O
of	O
,	O
but	O
also	O
(	O
trivially	O
)	O
isomorphic	O
to	O
a	O
subgroup	O
of	O
.	O
</s>
<s>
The	O
problem	O
of	O
finding	O
the	O
minimal-order	O
symmetric	B-Algorithm
group	I-Algorithm
into	O
which	O
a	O
given	O
group	O
embeds	O
is	O
rather	O
difficult	O
.	O
</s>
<s>
Alperin	O
and	O
Bell	O
note	O
that	O
"	O
in	O
general	O
the	O
fact	O
that	O
finite	O
groups	O
are	O
imbedded	O
in	O
symmetric	B-Algorithm
groups	I-Algorithm
has	O
not	O
influenced	O
the	O
methods	O
used	O
to	O
study	O
finite	O
groups	O
"	O
.	O
</s>
<s>
When	O
is	O
infinite	O
,	O
is	O
infinite	O
,	O
but	O
Cayley	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
still	O
applies	O
.	O
</s>
<s>
While	O
it	O
seems	O
elementary	O
enough	O
,	O
at	O
the	O
time	O
the	O
modern	O
definitions	O
did	O
not	O
exist	O
,	O
and	O
when	O
Cayley	O
introduced	O
what	O
are	O
now	O
called	O
groups	O
it	O
was	O
not	O
immediately	O
clear	O
that	O
this	O
was	O
equivalent	O
to	O
the	O
previously	O
known	O
groups	O
,	O
which	O
are	O
now	O
called	O
permutation	B-Algorithm
groups	O
.	O
</s>
<s>
Cayley	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
unifies	O
the	O
two	O
.	O
</s>
<s>
nonetheless	O
argues	O
that	O
the	O
standard	O
name	O
"	O
Cayley	B-Algorithm
's	I-Algorithm
Theorem	I-Algorithm
"	O
is	O
in	O
fact	O
appropriate	O
.	O
</s>
<s>
A	O
permutation	B-Algorithm
of	O
a	O
set	O
is	O
a	O
bijective	B-Algorithm
function	I-Algorithm
from	O
to	O
.	O
</s>
<s>
The	O
set	O
of	O
all	O
permutations	B-Algorithm
of	O
forms	O
a	O
group	O
under	O
function	B-Application
composition	I-Application
,	O
called	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
,	O
and	O
written	O
as	O
.	O
</s>
<s>
In	O
particular	O
,	O
taking	O
to	O
be	O
the	O
underlying	O
set	O
of	O
a	O
group	O
produces	O
a	O
symmetric	B-Algorithm
group	I-Algorithm
denoted	O
.	O
</s>
<s>
So	O
multiplication	O
by	O
g	O
acts	O
as	O
a	O
bijective	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
fg	O
is	O
a	O
permutation	B-Algorithm
of	O
G	O
,	O
and	O
so	O
is	O
a	O
member	O
of	O
Sym(G )	O
.	O
</s>
<s>
,	O
which	O
has	O
a	O
permutation	B-Algorithm
representation	O
,	O
say	O
.	O
</s>
<s>
The	O
identity	O
element	O
of	O
the	O
group	O
corresponds	O
to	O
the	O
identity	O
permutation	B-Algorithm
.	O
</s>
<s>
All	O
other	O
group	O
elements	O
correspond	O
to	O
derangements	O
:	O
permutations	B-Algorithm
that	O
do	O
not	O
leave	O
any	O
element	O
unchanged	O
.	O
</s>
<s>
Since	O
this	O
also	O
applies	O
for	O
powers	O
of	O
a	O
group	O
element	O
,	O
lower	O
than	O
the	O
order	O
of	O
that	O
element	O
,	O
each	O
element	O
corresponds	O
to	O
a	O
permutation	B-Algorithm
that	O
consists	O
of	O
cycles	O
all	O
of	O
the	O
same	O
length	O
:	O
this	O
length	O
is	O
the	O
order	O
of	O
that	O
element	O
.	O
</s>
<s>
Z2	O
=	O
 { 0 , 1 } 	O
with	O
addition	O
modulo	O
2	O
;	O
group	O
element	O
0	O
corresponds	O
to	O
the	O
identity	O
permutation	B-Algorithm
e	O
,	O
group	O
element	O
1	O
to	O
permutation	B-Algorithm
(	O
12	O
)	O
(	O
see	O
cycle	O
notation	O
)	O
.	O
</s>
<s>
0	O
+1	O
=	O
1	O
and	O
1+1	O
=	O
0	O
,	O
so	O
and	O
as	O
they	O
would	O
under	O
a	O
permutation	B-Algorithm
.	O
</s>
<s>
with	O
addition	O
modulo	O
3	O
;	O
group	O
element	O
0	O
corresponds	O
to	O
the	O
identity	O
permutation	B-Algorithm
e	O
,	O
group	O
element	O
1	O
to	O
permutation	B-Algorithm
(	O
123	O
)	O
,	O
and	O
group	O
element	O
2	O
to	O
permutation	B-Algorithm
(	O
132	O
)	O
.	O
</s>
<s>
S3	O
(	O
dihedral	O
group	O
of	O
order	O
6	O
)	O
is	O
the	O
group	O
of	O
all	O
permutations	B-Algorithm
of	O
3	O
objects	O
,	O
but	O
also	O
a	O
permutation	B-Algorithm
group	O
of	O
the	O
6	O
group	O
elements	O
,	O
and	O
the	O
latter	O
is	O
how	O
it	O
is	O
realized	O
by	O
its	O
regular	O
representation	O
.	O
</s>
