<s>
In	O
mathematics	O
,	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
an	O
object	O
X	O
is	O
the	O
group	O
consisting	O
of	O
automorphisms	O
of	O
X	O
under	O
composition	B-Application
of	O
morphisms	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
X	O
is	O
a	O
finite-dimensional	O
vector	O
space	O
,	O
then	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
X	O
is	O
the	O
group	O
of	O
invertible	O
linear	B-Architecture
transformations	I-Architecture
from	O
X	O
to	O
itself	O
(	O
the	O
general	O
linear	O
group	O
of	O
X	O
)	O
.	O
</s>
<s>
If	O
instead	O
X	O
is	O
a	O
group	O
,	O
then	O
its	O
automorphism	B-Algorithm
group	I-Algorithm
is	O
the	O
group	O
consisting	O
of	O
all	O
group	O
automorphisms	O
of	O
X	O
.	O
</s>
<s>
Especially	O
in	O
geometric	O
contexts	O
,	O
an	O
automorphism	B-Algorithm
group	I-Algorithm
is	O
also	O
called	O
a	O
symmetry	O
group	O
.	O
</s>
<s>
A	O
subgroup	O
of	O
an	O
automorphism	B-Algorithm
group	I-Algorithm
is	O
sometimes	O
called	O
a	O
transformation	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Automorphism	B-Algorithm
groups	I-Algorithm
are	O
studied	O
in	O
a	O
general	O
way	O
in	O
the	O
field	O
of	O
category	O
theory	O
.	O
</s>
<s>
If	O
X	O
is	O
a	O
set	O
with	O
no	O
additional	O
structure	O
,	O
then	O
any	O
bijection	B-Algorithm
from	O
X	O
to	O
itself	O
is	O
an	O
automorphism	O
,	O
and	O
hence	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
X	O
in	O
this	O
case	O
is	O
precisely	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
X	O
.	O
</s>
<s>
If	O
the	O
set	O
X	O
has	O
additional	O
structure	O
,	O
then	O
it	O
may	O
be	O
the	O
case	O
that	O
not	O
all	O
bijections	B-Algorithm
on	O
the	O
set	O
preserve	O
this	O
structure	O
,	O
in	O
which	O
case	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
will	O
be	O
a	O
subgroup	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
X	O
.	O
</s>
<s>
The	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
a	O
field	O
extension	O
is	O
the	O
group	O
consisting	O
of	O
field	O
automorphisms	O
of	O
L	O
that	O
fix	O
K	O
.	O
If	O
the	O
field	O
extension	O
is	O
Galois	O
,	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
is	O
called	O
the	O
Galois	O
group	O
of	O
the	O
field	O
extension	O
.	O
</s>
<s>
The	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
a	O
finite	O
cyclic	O
group	O
of	O
order	O
n	O
is	O
isomorphic	O
to	O
,	O
the	O
multiplicative	O
group	O
of	O
integers	O
modulo	O
n	O
,	O
with	O
the	O
isomorphism	O
given	O
by	O
.	O
</s>
<s>
The	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
a	O
finite-dimensional	O
real	O
Lie	O
algebra	O
has	O
the	O
structure	O
of	O
a	O
(	O
real	O
)	O
Lie	O
group	O
(	O
in	O
fact	O
,	O
it	O
is	O
even	O
a	O
linear	O
algebraic	O
group	O
:	O
see	O
below	O
)	O
.	O
</s>
<s>
If	O
G	O
is	O
a	O
Lie	O
group	O
with	O
Lie	O
algebra	O
,	O
then	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
G	O
has	O
a	O
structure	O
of	O
a	O
Lie	O
group	O
induced	O
from	O
that	O
on	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
.	O
</s>
<s>
If	O
G	O
is	O
a	O
group	O
acting	O
on	O
a	O
set	O
X	O
,	O
the	O
action	O
amounts	O
to	O
a	O
group	O
homomorphism	O
from	O
G	O
to	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
X	O
and	O
conversely	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
X	O
is	O
a	O
vector	O
space	O
,	O
then	O
a	O
group	O
action	O
of	O
G	O
on	O
X	O
is	O
a	O
group	O
representation	O
of	O
the	O
group	O
G	O
,	O
representing	O
G	O
as	O
a	O
group	O
of	O
linear	B-Architecture
transformations	I-Architecture
(	O
automorphisms	O
)	O
of	O
X	O
;	O
these	O
representations	O
are	O
the	O
main	O
object	O
of	O
study	O
in	O
the	O
field	O
of	O
representation	O
theory	O
.	O
</s>
<s>
Here	O
are	O
some	O
other	O
facts	O
about	O
automorphism	B-Algorithm
groups	I-Algorithm
:	O
</s>
<s>
Let	O
be	O
two	O
finite	O
sets	O
of	O
the	O
same	O
cardinality	B-Application
and	O
the	O
set	O
of	O
all	O
bijections	B-Algorithm
.	O
</s>
<s>
Then	O
,	O
which	O
is	O
a	O
symmetric	B-Algorithm
group	I-Algorithm
(	O
see	O
above	O
)	O
,	O
acts	O
on	O
from	O
the	O
left	O
freely	O
and	O
transitively	O
;	O
that	O
is	O
to	O
say	O
,	O
is	O
a	O
torsor	O
for	O
(	O
cf	O
.	O
</s>
<s>
Let	O
P	O
be	O
a	O
finitely	O
generated	O
projective	O
module	O
over	O
a	O
ring	O
R	O
.	O
Then	O
there	O
is	O
an	O
embedding	O
,	O
unique	O
up	O
to	O
inner	B-Algorithm
automorphisms	I-Algorithm
.	O
</s>
<s>
Automorphism	B-Algorithm
groups	I-Algorithm
appear	O
very	O
naturally	O
in	O
category	O
theory	O
.	O
</s>
<s>
If	O
X	O
is	O
an	O
object	O
in	O
a	O
category	O
,	O
then	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
X	O
is	O
the	O
group	O
consisting	O
of	O
all	O
the	O
invertible	O
morphisms	O
from	O
X	O
to	O
itself	O
.	O
</s>
<s>
If	O
and	O
are	O
objects	O
in	O
categories	O
and	O
,	O
and	O
if	O
is	O
a	O
functor	B-Language
mapping	O
to	O
,	O
then	O
induces	O
a	O
group	O
homomorphism	O
,	O
as	O
it	O
maps	O
invertible	O
morphisms	O
to	O
invertible	O
morphisms	O
.	O
</s>
<s>
In	O
particular	O
,	O
if	O
G	O
is	O
a	O
group	O
viewed	O
as	O
a	O
category	O
with	O
a	O
single	O
object	O
*	O
or	O
,	O
more	O
generally	O
,	O
if	O
G	O
is	O
a	O
groupoid	O
,	O
then	O
each	O
functor	B-Language
,	O
C	O
a	O
category	O
,	O
is	O
called	O
an	O
action	O
or	O
a	O
representation	O
of	O
G	O
on	O
the	O
object	O
,	O
or	O
the	O
objects	O
.	O
</s>
<s>
The	O
unit	O
group	O
of	O
is	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
When	O
a	O
basis	O
on	O
M	O
is	O
chosen	O
,	O
is	O
the	O
space	O
of	O
square	B-Algorithm
matrices	I-Algorithm
and	O
is	O
the	O
zero	O
set	O
of	O
some	O
polynomial	O
equations	O
,	O
and	O
the	O
invertibility	O
is	O
again	O
described	O
by	O
polynomials	O
.	O
</s>
<s>
Now	O
base	O
extensions	O
applied	O
to	O
the	O
above	O
discussion	O
determines	O
a	O
functor	B-Language
:	O
namely	O
,	O
for	O
each	O
commutative	O
ring	O
R	O
over	O
k	O
,	O
consider	O
the	O
R-linear	O
maps	O
preserving	O
the	O
algebraic	O
structure	O
:	O
denote	O
it	O
by	O
.	O
</s>
<s>
Then	O
the	O
unit	O
group	O
of	O
the	O
matrix	O
ring	O
over	O
R	O
is	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
and	O
is	O
a	O
group	O
functor	B-Language
:	O
a	O
functor	B-Language
from	O
the	O
category	O
of	O
commutative	O
rings	O
over	O
k	O
to	O
the	O
category	O
of	O
groups	O
.	O
</s>
<s>
Even	O
better	O
,	O
it	O
is	O
represented	O
by	O
a	O
scheme	O
(	O
since	O
the	O
automorphism	B-Algorithm
groups	I-Algorithm
are	O
defined	O
by	O
polynomials	O
)	O
:	O
this	O
scheme	O
is	O
called	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
scheme	O
and	O
is	O
denoted	O
by	O
.	O
</s>
<s>
In	O
general	O
,	O
however	O
,	O
an	O
automorphism	B-Algorithm
group	I-Algorithm
functor	B-Language
may	O
not	O
be	O
represented	O
by	O
a	O
scheme	O
.	O
</s>
