<s>
In	O
abstract	O
algebra	O
,	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
defined	O
over	O
any	O
set	O
is	O
the	O
group	O
whose	O
elements	O
are	O
all	O
the	O
bijections	B-Algorithm
from	O
the	O
set	O
to	O
itself	O
,	O
and	O
whose	O
group	O
operation	O
is	O
the	O
composition	B-Application
of	I-Application
functions	I-Application
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
finite	O
symmetric	B-Algorithm
group	I-Algorithm
defined	O
over	O
a	O
finite	O
set	O
of	O
symbols	O
consists	O
of	O
the	O
permutations	B-Algorithm
that	O
can	O
be	O
performed	O
on	O
the	O
symbols	O
.	O
</s>
<s>
Since	O
there	O
are	O
(	O
factorial	O
)	O
such	O
permutation	B-Algorithm
operations	O
,	O
the	O
order	O
(	O
number	O
of	O
elements	O
)	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
is	O
.	O
</s>
<s>
Although	O
symmetric	B-Algorithm
groups	I-Algorithm
can	O
be	O
defined	O
on	O
infinite	O
sets	O
,	O
this	O
article	O
focuses	O
on	O
the	O
finite	O
symmetric	B-Algorithm
groups	I-Algorithm
:	O
their	O
applications	O
,	O
their	O
elements	O
,	O
their	O
conjugacy	O
classes	O
,	O
a	O
finite	O
presentation	O
,	O
their	O
subgroups	O
,	O
their	O
automorphism	B-Algorithm
groups	I-Algorithm
,	O
and	O
their	O
representation	O
theory	O
.	O
</s>
<s>
For	O
the	O
remainder	O
of	O
this	O
article	O
,	O
"	O
symmetric	B-Algorithm
group	I-Algorithm
"	O
will	O
mean	O
a	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
a	O
finite	O
set	O
.	O
</s>
<s>
The	O
symmetric	B-Algorithm
group	I-Algorithm
is	O
important	O
to	O
diverse	O
areas	O
of	O
mathematics	O
such	O
as	O
Galois	O
theory	O
,	O
invariant	O
theory	O
,	O
the	O
representation	O
theory	O
of	O
Lie	O
groups	O
,	O
and	O
combinatorics	O
.	O
</s>
<s>
Cayley	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
states	O
that	O
every	O
group	O
is	O
isomorphic	O
to	O
a	O
subgroup	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
(	O
the	O
underlying	O
set	O
of	O
)	O
.	O
</s>
<s>
The	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
a	O
finite	O
set	O
is	O
the	O
group	O
whose	O
elements	O
are	O
all	O
bijective	B-Algorithm
functions	I-Algorithm
from	O
to	O
and	O
whose	O
group	O
operation	O
is	O
that	O
of	O
function	B-Application
composition	I-Application
.	O
</s>
<s>
For	O
finite	O
sets	O
,	O
"	O
permutations	B-Algorithm
"	O
and	O
"	O
bijective	B-Algorithm
functions	I-Algorithm
"	O
refer	O
to	O
the	O
same	O
operation	O
,	O
namely	O
rearrangement	O
.	O
</s>
<s>
The	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
degree	O
is	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
the	O
set	O
.	O
</s>
<s>
The	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
a	O
set	O
is	O
denoted	O
in	O
various	O
ways	O
,	O
including	O
,	O
,	O
,	O
,	O
and	O
.	O
</s>
<s>
Symmetric	B-Algorithm
groups	I-Algorithm
on	O
infinite	O
sets	O
behave	O
quite	O
differently	O
from	O
symmetric	B-Algorithm
groups	I-Algorithm
on	O
finite	O
sets	O
,	O
and	O
are	O
discussed	O
in	O
,	O
,	O
and	O
.	O
</s>
<s>
The	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
a	O
set	O
of	O
elements	O
has	O
order	O
(	O
the	O
factorial	O
of	O
)	O
.	O
</s>
<s>
For	O
and	O
(	O
the	O
empty	O
set	O
and	O
the	O
singleton	O
set	O
)	O
,	O
the	O
symmetric	B-Algorithm
groups	I-Algorithm
are	O
trivial	O
(	O
they	O
have	O
order	O
)	O
.	O
</s>
<s>
The	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
a	O
set	O
of	O
size	O
n	O
is	O
the	O
Galois	O
group	O
of	O
the	O
general	O
polynomial	O
of	O
degree	O
n	O
and	O
plays	O
an	O
important	O
role	O
in	O
Galois	O
theory	O
.	O
</s>
<s>
In	O
invariant	O
theory	O
,	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
acts	O
on	O
the	O
variables	O
of	O
a	O
multi-variate	O
function	O
,	O
and	O
the	O
functions	O
left	O
invariant	O
are	O
the	O
so-called	O
symmetric	O
functions	O
.	O
</s>
<s>
In	O
the	O
representation	O
theory	O
of	O
Lie	O
groups	O
,	O
the	O
representation	O
theory	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
plays	O
a	O
fundamental	O
role	O
through	O
the	O
ideas	O
of	O
Schur	O
functors	O
.	O
</s>
<s>
In	O
the	O
theory	O
of	O
Coxeter	B-Algorithm
groups	I-Algorithm
,	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
is	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
of	O
type	O
An	O
and	O
occurs	O
as	O
the	O
Weyl	O
group	O
of	O
the	O
general	O
linear	O
group	O
.	O
</s>
<s>
In	O
combinatorics	O
,	O
the	O
symmetric	B-Algorithm
groups	I-Algorithm
,	O
their	O
elements	O
(	O
permutations	B-Algorithm
)	O
,	O
and	O
their	O
representations	O
provide	O
a	O
rich	O
source	O
of	O
problems	O
involving	O
Young	O
tableaux	O
,	O
plactic	O
monoids	O
,	O
and	O
the	O
Bruhat	O
order	O
.	O
</s>
<s>
Subgroups	O
of	O
symmetric	B-Algorithm
groups	I-Algorithm
are	O
called	O
permutation	B-Algorithm
groups	I-Algorithm
and	O
are	O
widely	O
studied	O
because	O
of	O
their	O
importance	O
in	O
understanding	O
group	O
actions	O
,	O
homogeneous	O
spaces	O
,	O
and	O
automorphism	B-Algorithm
groups	I-Algorithm
of	O
graphs	O
,	O
such	O
as	O
the	O
Higman	O
–	O
Sims	O
group	O
and	O
the	O
Higman	O
–	O
Sims	O
graph	O
.	O
</s>
<s>
The	O
elements	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
a	O
set	O
X	O
are	O
the	O
permutations	B-Algorithm
of	O
X	O
.	O
</s>
<s>
The	O
group	O
operation	O
in	O
a	O
symmetric	B-Algorithm
group	I-Algorithm
is	O
function	B-Application
composition	I-Application
,	O
denoted	O
by	O
the	O
symbol	O
∘	O
or	O
simply	O
by	O
just	O
a	O
composition	O
of	O
the	O
permutations	B-Algorithm
.	O
</s>
<s>
The	O
composition	O
of	O
permutations	B-Algorithm
f	O
and	O
g	O
,	O
pronounced	O
"	O
f	O
of	O
g	O
"	O
,	O
maps	O
any	O
element	O
x	O
of	O
X	O
to	O
f(g(x )	O
)	O
.	O
</s>
<s>
Concretely	O
,	O
let	O
(	O
see	O
permutation	B-Algorithm
for	O
an	O
explanation	O
of	O
notation	O
)	O
:	O
</s>
<s>
A	O
cycle	B-Algorithm
of	O
length	O
,	O
taken	O
to	O
the	O
kth	O
power	O
,	O
will	O
decompose	O
into	O
k	O
cycles	O
of	O
length	O
m	O
:	O
For	O
example	O
,	O
(	O
,	O
)	O
,	O
</s>
<s>
To	O
check	O
that	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
a	O
set	O
X	O
is	O
indeed	O
a	O
group	O
,	O
it	O
is	O
necessary	O
to	O
verify	O
the	O
group	O
axioms	O
of	O
closure	O
,	O
associativity	O
,	O
identity	O
,	O
and	O
inverses	O
.	O
</s>
<s>
The	O
operation	O
of	O
function	B-Application
composition	I-Application
is	O
closed	O
in	O
the	O
set	O
of	O
permutations	B-Algorithm
of	O
the	O
given	O
set	O
X	O
.	O
</s>
<s>
Function	B-Application
composition	I-Application
is	O
always	O
associative	O
.	O
</s>
<s>
The	O
trivial	O
bijection	B-Algorithm
that	O
assigns	O
each	O
element	O
of	O
X	O
to	O
itself	O
serves	O
as	O
an	O
identity	O
for	O
the	O
group	O
.	O
</s>
<s>
Every	O
bijection	B-Algorithm
has	O
an	O
inverse	O
function	O
that	O
undoes	O
its	O
action	O
,	O
and	O
thus	O
each	O
element	O
of	O
a	O
symmetric	B-Algorithm
group	I-Algorithm
does	O
have	O
an	O
inverse	O
which	O
is	O
a	O
permutation	B-Algorithm
too	O
.	O
</s>
<s>
A	O
transposition	O
is	O
a	O
permutation	B-Algorithm
which	O
exchanges	O
two	O
elements	O
and	O
keeps	O
all	O
others	O
fixed	O
;	O
for	O
example	O
(	O
1	O
3	O
)	O
is	O
a	O
transposition	O
.	O
</s>
<s>
Every	O
permutation	B-Algorithm
can	O
be	O
written	O
as	O
a	O
product	O
of	O
transpositions	O
;	O
for	O
instance	O
,	O
the	O
permutation	B-Algorithm
g	O
from	O
above	O
can	O
be	O
written	O
as	O
g	O
=	O
(	O
1	O
2	O
)	O
(	O
2	O
5	O
)	O
(	O
3	O
4	O
)	O
.	O
</s>
<s>
Since	O
g	O
can	O
be	O
written	O
as	O
a	O
product	O
of	O
an	O
odd	O
number	O
of	O
transpositions	O
,	O
it	O
is	O
then	O
called	O
an	O
odd	O
permutation	B-Algorithm
,	O
whereas	O
f	O
is	O
an	O
even	O
permutation	B-Algorithm
.	O
</s>
<s>
The	O
representation	O
of	O
a	O
permutation	B-Algorithm
as	O
a	O
product	O
of	O
transpositions	O
is	O
not	O
unique	O
;	O
however	O
,	O
the	O
number	O
of	O
transpositions	O
needed	O
to	O
represent	O
a	O
given	O
permutation	B-Algorithm
is	O
either	O
always	O
even	O
or	O
always	O
odd	O
.	O
</s>
<s>
There	O
are	O
several	O
short	O
proofs	O
of	O
the	O
invariance	O
of	O
this	O
parity	O
of	O
a	O
permutation	B-Algorithm
.	O
</s>
<s>
The	O
product	O
of	O
two	O
even	O
permutations	B-Algorithm
is	O
even	O
,	O
the	O
product	O
of	O
two	O
odd	O
permutations	B-Algorithm
is	O
even	O
,	O
and	O
all	O
other	O
products	O
are	O
odd	O
.	O
</s>
<s>
Thus	O
we	O
can	O
define	O
the	O
sign	O
of	O
a	O
permutation	B-Algorithm
:	O
</s>
<s>
The	O
kernel	O
of	O
this	O
homomorphism	O
,	O
that	O
is	O
,	O
the	O
set	O
of	O
all	O
even	O
permutations	B-Algorithm
,	O
is	O
called	O
the	O
alternating	B-Algorithm
group	I-Algorithm
An	O
.	O
</s>
<s>
Furthermore	O
,	O
every	O
permutation	B-Algorithm
can	O
be	O
written	O
as	O
a	O
product	O
of	O
adjacent	O
transpositions	O
,	O
that	O
is	O
,	O
transpositions	O
of	O
the	O
form	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
permutation	B-Algorithm
g	O
from	O
above	O
can	O
also	O
be	O
written	O
as	O
.	O
</s>
<s>
The	O
sorting	O
algorithm	O
bubble	B-Algorithm
sort	I-Algorithm
is	O
an	O
application	O
of	O
this	O
fact	O
.	O
</s>
<s>
The	O
representation	O
of	O
a	O
permutation	B-Algorithm
as	O
a	O
product	O
of	O
adjacent	O
transpositions	O
is	O
also	O
not	O
unique	O
.	O
</s>
<s>
A	O
cycle	B-Algorithm
of	O
length	O
k	O
is	O
a	O
permutation	B-Algorithm
f	O
for	O
which	O
there	O
exists	O
an	O
element	O
x	O
in	O
{	O
1	O
,	O
...	O
,	O
n}	O
such	O
that	O
x	O
,	O
f(x )	O
,	O
f2(x )	O
,	O
...	O
,	O
fk(x )	O
=	O
x	O
are	O
the	O
only	O
elements	O
moved	O
by	O
f	O
;	O
it	O
is	O
required	O
that	O
since	O
with	O
the	O
element	O
x	O
itself	O
would	O
not	O
be	O
moved	O
either	O
.	O
</s>
<s>
is	O
a	O
cycle	B-Algorithm
of	O
length	O
three	O
,	O
since	O
,	O
and	O
,	O
leaving	O
2	O
and	O
5	O
untouched	O
.	O
</s>
<s>
We	O
denote	O
such	O
a	O
cycle	B-Algorithm
by	O
,	O
but	O
it	O
could	O
equally	O
well	O
be	O
written	O
or	O
by	O
starting	O
at	O
a	O
different	O
point	O
.	O
</s>
<s>
The	O
order	O
of	O
a	O
cycle	B-Algorithm
is	O
equal	O
to	O
its	O
length	O
.	O
</s>
<s>
Every	O
element	O
of	O
Sn	O
can	O
be	O
written	O
as	O
a	O
product	O
of	O
disjoint	O
cycles	O
;	O
this	O
representation	O
is	O
unique	O
up	O
to	O
the	O
order	O
of	O
the	O
factors	O
,	O
and	O
the	O
freedom	O
present	O
in	O
representing	O
each	O
individual	O
cycle	B-Algorithm
by	O
choosing	O
its	O
starting	O
point	O
.	O
</s>
<s>
Cycles	O
admit	O
the	O
following	O
conjugation	O
property	O
with	O
any	O
permutation	B-Algorithm
,	O
this	O
property	O
is	O
often	O
used	O
to	O
obtain	O
its	O
generators	O
and	O
relations	O
.	O
</s>
<s>
Certain	O
elements	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
{	O
1	O
,	O
2	O
,	O
...	O
,	O
n}	O
are	O
of	O
particular	O
interest	O
(	O
these	O
can	O
be	O
generalized	O
to	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
any	O
finite	O
totally	O
ordered	O
set	O
,	O
but	O
not	O
to	O
that	O
of	O
an	O
unordered	O
set	O
)	O
.	O
</s>
<s>
longest	O
element	O
in	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
with	O
respect	O
to	O
generating	O
set	O
consisting	O
of	O
the	O
adjacent	O
transpositions	O
,	O
.	O
</s>
<s>
In	O
S2n	O
,	O
the	O
perfect	O
shuffle	O
is	O
the	O
permutation	B-Algorithm
that	O
splits	O
the	O
set	O
into	O
2	O
piles	O
and	O
interleaves	O
them	O
.	O
</s>
<s>
The	O
conjugacy	O
classes	O
of	O
Sn	O
correspond	O
to	O
the	O
cycle	B-Algorithm
types	O
of	O
permutations	B-Algorithm
;	O
that	O
is	O
,	O
two	O
elements	O
of	O
Sn	O
are	O
conjugate	O
in	O
Sn	O
if	O
and	O
only	O
if	O
they	O
consist	O
of	O
the	O
same	O
number	O
of	O
disjoint	O
cycles	O
of	O
the	O
same	O
lengths	O
.	O
</s>
<s>
A	O
conjugating	O
element	O
of	O
Sn	O
can	O
be	O
constructed	O
in	O
"	O
two	O
line	O
notation	O
"	O
by	O
placing	O
the	O
"	O
cycle	B-Algorithm
notations	O
"	O
of	O
the	O
two	O
conjugate	O
permutations	B-Algorithm
on	O
top	O
of	O
one	O
another	O
.	O
</s>
<s>
This	O
permutation	B-Algorithm
then	O
relates	O
(	O
1	O
2	O
3	O
)	O
(	O
4	O
5	O
)	O
and	O
(	O
1	O
4	O
3	O
)	O
(	O
2	O
5	O
)	O
via	O
conjugation	O
,	O
that	O
is	O
,	O
</s>
<s>
It	O
is	O
clear	O
that	O
such	O
a	O
permutation	B-Algorithm
is	O
not	O
unique	O
.	O
</s>
<s>
Conjugacy	O
classes	O
of	O
correspond	O
to	O
integer	O
partitions	O
of	O
:	O
to	O
the	O
partition	O
with	O
and	O
,	O
is	O
associated	O
the	O
set	O
of	O
permutations	B-Algorithm
with	O
cycles	O
of	O
lengths	O
.	O
</s>
<s>
Then	O
is	O
a	O
conjugacy	O
class	O
of	O
,	O
whose	O
elements	O
are	O
said	O
to	O
be	O
of	O
cycle-type	O
.	O
</s>
<s>
The	O
low-degree	O
symmetric	B-Algorithm
groups	I-Algorithm
have	O
simpler	O
and	O
exceptional	O
structure	O
,	O
and	O
often	O
must	O
be	O
treated	O
separately	O
.	O
</s>
<s>
S0	O
and	O
S1	O
The	O
symmetric	B-Algorithm
groups	I-Algorithm
on	O
the	O
empty	O
set	O
and	O
the	O
singleton	O
set	O
are	O
trivial	O
,	O
which	O
corresponds	O
to	O
.	O
</s>
<s>
In	O
this	O
case	O
the	O
alternating	B-Algorithm
group	I-Algorithm
agrees	O
with	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
rather	O
than	O
being	O
an	O
index	O
2	O
subgroup	O
,	O
and	O
the	O
sign	O
map	O
is	O
trivial	O
.	O
</s>
<s>
S2	O
This	O
group	O
consists	O
of	O
exactly	O
two	O
elements	O
:	O
the	O
identity	O
and	O
the	O
permutation	B-Algorithm
swapping	O
the	O
two	O
points	O
.	O
</s>
<s>
In	O
invariant	O
theory	O
,	O
the	O
representation	O
theory	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
two	O
points	O
is	O
quite	O
simple	O
and	O
is	O
seen	O
as	O
writing	O
a	O
function	O
of	O
two	O
variables	O
as	O
a	O
sum	O
of	O
its	O
symmetric	O
and	O
anti-symmetric	O
parts	O
:	O
Setting	O
,	O
and	O
,	O
one	O
gets	O
that	O
.	O
</s>
<s>
S3	O
S3	O
is	O
the	O
first	O
nonabelian	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
In	O
Galois	O
theory	O
,	O
the	O
sign	O
map	O
from	O
S3	O
to	O
S2	O
corresponds	O
to	O
the	O
resolving	O
quadratic	O
for	O
a	O
cubic	O
polynomial	O
,	O
as	O
discovered	O
by	O
Gerolamo	O
Cardano	O
,	O
while	O
the	O
A3	O
kernel	O
corresponds	O
to	O
the	O
use	O
of	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
of	O
order	O
3	O
in	O
the	O
solution	O
,	O
in	O
the	O
form	O
of	O
Lagrange	O
resolvents	O
.	O
</s>
<s>
S4	O
The	O
group	O
S4	O
is	O
isomorphic	O
to	O
the	O
group	O
of	O
proper	O
rotations	O
about	O
opposite	O
faces	O
,	O
opposite	O
diagonals	O
and	O
opposite	O
edges	O
,	O
9	O
,	O
8	O
and	O
6	O
permutations	B-Algorithm
,	O
of	O
the	O
cube	B-Application
.	O
</s>
<s>
The	O
map	O
from	O
S4	O
to	O
S3	O
also	O
yields	O
a	O
2-dimensional	O
irreducible	O
representation	O
,	O
which	O
is	O
an	O
irreducible	O
representation	O
of	O
a	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
degree	O
n	O
of	O
dimension	O
below	O
,	O
which	O
only	O
occurs	O
for	O
.	O
</s>
<s>
S5	O
S5	O
is	O
the	O
first	O
non-solvable	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
This	O
yields	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
of	O
S6	O
,	O
discussed	O
below	O
,	O
and	O
corresponds	O
to	O
the	O
resolvent	O
sextic	O
of	O
a	O
quintic	O
.	O
</s>
<s>
S6	O
Unlike	O
all	O
other	O
symmetric	B-Algorithm
groups	I-Algorithm
,	O
S6	O
,	O
has	O
an	O
outer	B-Algorithm
automorphism	I-Algorithm
.	O
</s>
<s>
The	O
resolvent	O
of	O
a	O
quintic	O
is	O
of	O
degree	O
6	O
—	O
this	O
corresponds	O
to	O
an	O
exotic	O
inclusion	O
map	O
as	O
a	O
transitive	O
subgroup	O
(	O
the	O
obvious	O
inclusion	O
map	O
fixes	O
a	O
point	O
and	O
thus	O
is	O
not	O
transitive	O
)	O
and	O
,	O
while	O
this	O
map	O
does	O
not	O
make	O
the	O
general	O
quintic	O
solvable	O
,	O
it	O
yields	O
the	O
exotic	O
outer	B-Algorithm
automorphism	I-Algorithm
of	O
S6	O
—	O
see	O
Automorphisms	O
of	O
the	O
symmetric	O
and	O
alternating	B-Algorithm
groups	I-Algorithm
for	O
details	O
.	O
</s>
<s>
Note	O
that	O
while	O
A6	O
and	O
A7	O
have	O
an	O
exceptional	O
Schur	O
multiplier	O
(	O
a	O
triple	B-Algorithm
cover	I-Algorithm
)	O
and	O
that	O
these	O
extend	O
to	O
triple	O
covers	O
of	O
S6	O
and	O
S7	O
,	O
these	O
do	O
not	O
correspond	O
to	O
exceptional	O
Schur	O
multipliers	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Other	O
than	O
the	O
trivial	O
map	O
and	O
the	O
sign	O
map	O
,	O
the	O
most	O
notable	O
homomorphisms	O
between	O
symmetric	B-Algorithm
groups	I-Algorithm
,	O
in	O
order	O
of	O
relative	O
dimension	O
,	O
are	O
:	O
</s>
<s>
(	O
or	O
rather	O
,	O
a	O
class	O
of	O
such	O
maps	O
up	O
to	O
inner	B-Algorithm
automorphism	I-Algorithm
)	O
corresponding	O
to	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
of	O
S6	O
.	O
</s>
<s>
as	O
a	O
transitive	O
subgroup	O
,	O
yielding	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
of	O
S6	O
as	O
discussed	O
above	O
.	O
</s>
<s>
For	O
,	O
the	O
alternating	B-Algorithm
group	I-Algorithm
An	O
is	O
simple	O
,	O
and	O
the	O
induced	O
quotient	O
is	O
the	O
sign	O
map	O
:	O
which	O
is	O
split	O
by	O
taking	O
a	O
transposition	O
of	O
two	O
elements	O
.	O
</s>
<s>
Sn	O
acts	O
on	O
its	O
subgroup	O
An	O
by	O
conjugation	O
,	O
and	O
for	O
,	O
Sn	O
is	O
the	O
full	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
An	O
:	O
Aut(An )	O
≅	O
Sn	O
.	O
</s>
<s>
Conjugation	O
by	O
even	O
elements	O
are	O
inner	B-Algorithm
automorphisms	I-Algorithm
of	O
An	O
while	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
of	O
An	O
of	O
order	O
2	O
corresponds	O
to	O
conjugation	O
by	O
an	O
odd	O
element	O
.	O
</s>
<s>
For	O
,	O
there	O
is	O
an	O
exceptional	O
outer	B-Algorithm
automorphism	I-Algorithm
of	O
An	O
so	O
Sn	O
is	O
not	O
the	O
full	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
An	O
.	O
</s>
<s>
Conversely	O
,	O
for	O
,	O
Sn	O
has	O
no	O
outer	B-Algorithm
automorphisms	I-Algorithm
,	O
and	O
for	O
it	O
has	O
no	O
center	O
,	O
so	O
for	O
it	O
is	O
a	O
complete	O
group	O
,	O
as	O
discussed	O
in	O
automorphism	B-Algorithm
group	I-Algorithm
,	O
below	O
.	O
</s>
<s>
Sn	O
can	O
be	O
embedded	O
into	O
An+2	O
by	O
appending	O
the	O
transposition	O
to	O
all	O
odd	O
permutations	B-Algorithm
,	O
while	O
embedding	O
into	O
An+1	O
is	O
impossible	O
for	O
.	O
</s>
<s>
The	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
letters	O
is	O
generated	O
by	O
the	O
adjacent	O
transpositions	O
that	O
swap	O
and	O
.	O
</s>
<s>
where	O
1	O
represents	O
the	O
identity	O
permutation	B-Algorithm
.	O
</s>
<s>
This	O
representation	O
endows	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
with	O
the	O
structure	O
of	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
(	O
and	O
so	O
also	O
a	O
reflection	B-Algorithm
group	I-Algorithm
)	O
.	O
</s>
<s>
Other	O
possible	O
generating	O
sets	O
include	O
the	O
set	O
of	O
transpositions	O
that	O
swap	O
and	O
for	O
,	O
and	O
a	O
set	O
containing	O
any	O
-cycle	O
and	O
a	O
-cycle	O
of	O
adjacent	O
elements	O
in	O
the	O
-cycle	O
.	O
</s>
<s>
A	O
subgroup	O
of	O
a	O
symmetric	B-Algorithm
group	I-Algorithm
is	O
called	O
a	O
permutation	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
normal	O
subgroups	O
of	O
the	O
finite	O
symmetric	B-Algorithm
groups	I-Algorithm
are	O
well	O
understood	O
.	O
</s>
<s>
The	O
alternating	B-Algorithm
group	I-Algorithm
of	O
degree	O
n	O
is	O
always	O
a	O
normal	O
subgroup	O
,	O
a	O
proper	O
one	O
for	O
and	O
nontrivial	O
for	O
;	O
for	O
it	O
is	O
in	O
fact	O
the	O
only	O
nontrivial	O
proper	O
normal	O
subgroup	O
of	O
Sn	O
,	O
except	O
when	O
where	O
there	O
is	O
one	O
additional	O
such	O
normal	O
subgroup	O
,	O
which	O
is	O
isomorphic	O
to	O
the	O
Klein	O
four	O
group	O
.	O
</s>
<s>
The	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
an	O
infinite	O
set	O
does	O
not	O
have	O
a	O
subgroup	O
of	O
index	O
2	O
,	O
as	O
Vitali	O
(	O
1915	O
)	O
proved	O
that	O
each	O
permutation	B-Algorithm
can	O
be	O
written	O
as	O
a	O
product	O
of	O
three	O
squares	O
.	O
</s>
<s>
However	O
it	O
contains	O
the	O
normal	O
subgroup	O
S	O
of	O
permutations	B-Algorithm
that	O
fix	O
all	O
but	O
finitely	O
many	O
elements	O
,	O
which	O
is	O
generated	O
by	O
transpositions	O
.	O
</s>
<s>
Since	O
A	O
is	O
even	O
a	O
characteristic	O
subgroup	O
of	O
S	O
,	O
it	O
is	O
also	O
a	O
normal	O
subgroup	O
of	O
the	O
full	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
the	O
infinite	O
set	O
.	O
</s>
<s>
The	O
groups	O
A	O
and	O
S	O
are	O
the	O
only	O
nontrivial	O
proper	O
normal	O
subgroups	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
a	O
countably	O
infinite	O
set	O
.	O
</s>
<s>
The	O
primitive	O
maximal	O
subgroups	O
are	O
more	O
difficult	O
to	O
identify	O
,	O
but	O
with	O
the	O
assistance	O
of	O
the	O
O'Nan–Scott	B-Algorithm
theorem	I-Algorithm
and	O
the	O
classification	O
of	O
finite	O
simple	O
groups	O
,	O
gave	O
a	O
fairly	O
satisfactory	O
description	O
of	O
the	O
maximal	O
subgroups	O
of	O
this	O
type	O
,	O
according	O
to	O
.	O
</s>
<s>
The	O
Sylow	O
subgroups	O
of	O
the	O
symmetric	B-Algorithm
groups	I-Algorithm
are	O
important	O
examples	O
of	O
p-groups	O
.	O
</s>
<s>
The	O
Sylow	O
p-subgroups	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
degree	O
p	O
are	O
just	O
the	O
cyclic	O
subgroups	O
generated	O
by	O
p-cycles	O
.	O
</s>
<s>
The	O
normalizer	O
therefore	O
has	O
order	O
and	O
is	O
known	O
as	O
a	O
Frobenius	B-Algorithm
group	I-Algorithm
(	O
especially	O
for	O
)	O
,	O
and	O
is	O
the	O
affine	O
general	O
linear	O
group	O
,	O
.	O
</s>
<s>
The	O
Sylow	O
p-subgroups	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
degree	O
p2	O
are	O
the	O
wreath	O
product	O
of	O
two	O
cyclic	O
groups	O
of	O
order	O
p	O
.	O
For	O
instance	O
,	O
when	O
,	O
a	O
Sylow	O
3-subgroup	O
of	O
Sym(9 )	O
is	O
generated	O
by	O
and	O
the	O
elements	O
,	O
and	O
every	O
element	O
of	O
the	O
Sylow	O
3-subgroup	O
has	O
the	O
form	O
for	O
.	O
</s>
<s>
The	O
Sylow	O
p-subgroups	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
degree	O
pn	O
are	O
sometimes	O
denoted	O
Wp(n )	O
,	O
and	O
using	O
this	O
notation	O
one	O
has	O
that	O
is	O
the	O
wreath	O
product	O
of	O
Wp(n )	O
and	O
Wp(1 )	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
Sylow	O
p-subgroups	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
degree	O
n	O
are	O
a	O
direct	O
product	O
of	O
ai	O
copies	O
of	O
Wp(i )	O
,	O
where	O
and	O
(	O
the	O
base	O
p	O
expansion	O
of	O
n	O
)	O
.	O
</s>
<s>
For	O
instance	O
,	O
,	O
the	O
dihedral	O
group	O
of	O
order	O
8	O
,	O
and	O
so	O
a	O
Sylow	O
2-subgroup	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
degree	O
7	O
is	O
generated	O
by	O
and	O
is	O
isomorphic	O
to	O
.	O
</s>
<s>
Cayley	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
states	O
that	O
every	O
group	O
G	O
is	O
isomorphic	O
to	O
a	O
subgroup	O
of	O
some	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
one	O
may	O
take	O
a	O
subgroup	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
the	O
elements	O
of	O
G	O
,	O
since	O
every	O
group	O
acts	O
on	O
itself	O
faithfully	O
by	O
(	O
left	O
or	O
right	O
)	O
multiplication	O
.	O
</s>
<s>
Cyclic	O
groups	O
are	O
those	O
that	O
are	O
generated	O
by	O
a	O
single	O
permutation	B-Algorithm
.	O
</s>
<s>
When	O
a	O
permutation	B-Algorithm
is	O
represented	O
in	O
cycle	B-Algorithm
notation	O
,	O
the	O
order	O
of	O
the	O
cyclic	O
subgroup	O
that	O
it	O
generates	O
is	O
the	O
least	O
common	O
multiple	O
of	O
the	O
lengths	O
of	O
its	O
cycles	O
.	O
</s>
<s>
For	O
example	O
,	O
in	O
S	O
,	O
one	O
cyclic	O
subgroup	O
of	O
order	O
5	O
is	O
generated	O
by	O
(	O
13254	O
)	O
,	O
whereas	O
the	O
largest	O
cyclic	O
subgroups	O
of	O
S	O
are	O
generated	O
by	O
elements	O
like	O
(	O
123	O
)	O
(	O
45	O
)	O
that	O
have	O
one	O
cycle	B-Algorithm
of	O
length	O
3	O
and	O
another	O
cycle	B-Algorithm
of	O
length	O
2	O
.	O
</s>
<s>
This	O
rules	O
out	O
many	O
groups	O
as	O
possible	O
subgroups	O
of	O
symmetric	B-Algorithm
groups	I-Algorithm
of	O
a	O
given	O
size	O
.	O
</s>
<s>
For	O
,	O
Sn	O
is	O
a	O
complete	O
group	O
:	O
its	O
center	O
and	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
are	O
both	O
trivial	O
.	O
</s>
<s>
For	O
,	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
is	O
trivial	O
,	O
but	O
S2	O
is	O
not	O
trivial	O
:	O
it	O
is	O
isomorphic	O
to	O
C2	O
,	O
which	O
is	O
abelian	O
,	O
and	O
hence	O
the	O
center	O
is	O
the	O
whole	O
group	O
.	O
</s>
<s>
For	O
,	O
it	O
has	O
an	O
outer	B-Algorithm
automorphism	I-Algorithm
of	O
order	O
2	O
:	O
,	O
and	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
is	O
a	O
semidirect	O
product	O
.	O
</s>
<s>
In	O
fact	O
,	O
for	O
any	O
set	O
X	O
of	O
cardinality	O
other	O
than	O
6	O
,	O
every	O
automorphism	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
X	O
is	O
inner	O
,	O
a	O
result	O
first	O
due	O
to	O
according	O
to	O
.	O
</s>
<s>
The	O
first	O
homology	O
group	O
is	O
the	O
abelianization	O
,	O
and	O
corresponds	O
to	O
the	O
sign	O
map	O
Sn	O
→	O
S2	O
which	O
is	O
the	O
abelianization	O
for	O
n	O
≥	O
2	O
;	O
for	O
n	O
<	O
2	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
is	O
trivial	O
.	O
</s>
<s>
This	O
was	O
computed	O
in	O
,	O
and	O
corresponds	O
to	O
the	O
double	B-Algorithm
cover	I-Algorithm
of	I-Algorithm
the	I-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
,	O
2	O
·	O
Sn	O
.	O
</s>
<s>
Note	O
that	O
the	O
exceptional	O
low-dimensional	O
homology	O
of	O
the	O
alternating	B-Algorithm
group	I-Algorithm
(	O
corresponding	O
to	O
non-trivial	O
abelianization	O
,	O
and	O
due	O
to	O
the	O
exceptional	O
3-fold	O
cover	O
)	O
does	O
not	O
change	O
the	O
homology	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
;	O
the	O
alternating	B-Algorithm
group	I-Algorithm
phenomena	O
do	O
yield	O
symmetric	B-Algorithm
group	I-Algorithm
phenomena	O
–	O
the	O
map	O
extends	O
to	O
and	O
the	O
triple	O
covers	O
of	O
A6	O
and	O
A7	O
extend	O
to	O
triple	O
covers	O
of	O
S6	O
and	O
S7	O
–	O
but	O
these	O
are	O
not	O
homological	O
–	O
the	O
map	O
does	O
not	O
change	O
the	O
abelianization	O
of	O
S4	O
,	O
and	O
the	O
triple	O
covers	O
do	O
not	O
correspond	O
to	O
homology	O
either	O
.	O
</s>
<s>
The	O
homology	O
of	O
the	O
infinite	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
is	O
computed	O
in	O
,	O
with	O
the	O
cohomology	O
algebra	O
forming	O
a	O
Hopf	O
algebra	O
.	O
</s>
<s>
The	O
representation	O
theory	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
is	O
a	O
particular	O
case	O
of	O
the	O
representation	O
theory	O
of	O
finite	O
groups	O
,	O
for	O
which	O
a	O
concrete	O
and	O
detailed	O
theory	O
can	O
be	O
obtained	O
.	O
</s>
<s>
The	O
symmetric	B-Algorithm
group	I-Algorithm
Sn	O
has	O
order	O
n	O
!	O
.	O
</s>
<s>
Each	O
such	O
irreducible	O
representation	O
can	O
be	O
realized	O
over	O
the	O
integers	O
(	O
every	O
permutation	B-Algorithm
acting	O
by	O
a	O
matrix	O
with	O
integer	O
coefficients	O
)	O
;	O
it	O
can	O
be	O
explicitly	O
constructed	O
by	O
computing	O
the	O
Young	O
symmetrizers	O
acting	O
on	O
a	O
space	O
generated	O
by	O
the	O
Young	O
tableaux	O
of	O
shape	O
given	O
by	O
the	O
Young	O
diagram	O
.	O
</s>
<s>
However	O
,	O
the	O
irreducible	O
representations	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
are	O
not	O
known	O
in	O
arbitrary	O
characteristic	O
.	O
</s>
<s>
The	O
determination	O
of	O
the	O
irreducible	O
modules	O
for	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
over	O
an	O
arbitrary	O
field	O
is	O
widely	O
regarded	O
as	O
one	O
of	O
the	O
most	O
important	O
open	O
problems	O
in	O
representation	O
theory	O
.	O
</s>
