<s>
In	O
mathematics	O
,	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
of	O
a	O
group	O
,	O
,	O
is	O
the	O
quotient	O
,	O
,	O
where	O
is	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
and	O
)	O
is	O
the	O
subgroup	O
consisting	O
of	O
inner	B-Algorithm
automorphisms	I-Algorithm
.	O
</s>
<s>
The	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
is	O
usually	O
denoted	O
.	O
</s>
<s>
An	O
automorphism	O
of	O
a	O
group	O
which	O
is	O
not	O
inner	O
is	O
called	O
an	O
outer	B-Algorithm
automorphism	I-Algorithm
.	O
</s>
<s>
The	O
cosets	O
of	O
with	O
respect	O
to	O
outer	B-Algorithm
automorphisms	I-Algorithm
are	O
then	O
the	O
elements	O
of	O
;	O
this	O
is	O
an	O
instance	O
of	O
the	O
fact	O
that	O
quotients	O
of	O
groups	O
are	O
not	O
,	O
in	O
general	O
,	O
(	O
isomorphic	O
to	O
)	O
subgroups	O
.	O
</s>
<s>
If	O
the	O
inner	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
is	O
trivial	O
(	O
when	O
a	O
group	O
is	O
abelian	O
)	O
,	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
and	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
are	O
naturally	O
identified	O
;	O
that	O
is	O
,	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
does	O
act	O
on	O
the	O
group	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
the	O
alternating	B-Algorithm
group	I-Algorithm
,	O
,	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
is	O
usually	O
the	O
group	O
of	O
order	O
2	O
,	O
with	O
exceptions	O
noted	O
below	O
.	O
</s>
<s>
Considering	O
as	O
a	O
subgroup	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
,	O
conjugation	O
by	O
any	O
odd	O
permutation	O
is	O
an	O
outer	B-Algorithm
automorphism	I-Algorithm
of	O
or	O
more	O
precisely	O
"	O
represents	O
the	O
class	O
of	O
the	O
(	O
non-trivial	O
)	O
outer	B-Algorithm
automorphism	I-Algorithm
of	O
"	O
,	O
but	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
does	O
not	O
correspond	O
to	O
conjugation	O
by	O
any	O
particular	O
odd	O
element	O
,	O
and	O
all	O
conjugations	O
by	O
odd	O
elements	O
are	O
equivalent	O
up	O
to	O
conjugation	O
by	O
an	O
even	O
element	O
.	O
</s>
<s>
The	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
is	O
dual	O
to	O
the	O
center	O
in	O
the	O
following	O
sense	O
:	O
conjugation	O
by	O
an	O
element	O
of	O
is	O
an	O
automorphism	O
,	O
yielding	O
a	O
map	O
.	O
</s>
<s>
The	O
kernel	O
of	O
the	O
conjugation	B-Algorithm
map	I-Algorithm
is	O
the	O
center	O
,	O
while	O
the	O
cokernel	O
is	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
(	O
and	O
the	O
image	O
is	O
the	O
inner	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
)	O
.	O
</s>
<s>
The	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
of	O
a	O
group	O
acts	O
on	O
conjugacy	O
classes	O
,	O
and	O
accordingly	O
on	O
the	O
character	O
table	O
.	O
</s>
<s>
See	O
details	O
at	O
character	O
table	O
:	O
outer	B-Algorithm
automorphisms	I-Algorithm
.	O
</s>
<s>
The	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
is	O
important	O
in	O
the	O
topology	B-Architecture
of	O
surfaces	O
because	O
there	O
is	O
a	O
connection	O
provided	O
by	O
the	O
Dehn	O
–	O
Nielsen	O
theorem	O
:	O
the	O
extended	O
mapping	O
class	O
group	O
of	O
the	O
surface	O
is	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
of	O
its	O
fundamental	O
group	O
.	O
</s>
<s>
For	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
groups	I-Algorithm
of	O
all	O
finite	O
simple	O
groups	O
see	O
the	O
list	O
of	O
finite	O
simple	O
groups	O
.	O
</s>
<s>
Sporadic	O
simple	O
groups	O
and	O
alternating	B-Algorithm
groups	I-Algorithm
(	O
other	O
than	O
the	O
alternating	B-Algorithm
group	I-Algorithm
,	O
;	O
see	O
below	O
)	O
all	O
have	O
outer	B-Algorithm
automorphism	I-Algorithm
groups	I-Algorithm
of	O
order	O
1	O
or	O
2	O
.	O
</s>
<s>
The	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
of	O
a	O
finite	O
simple	O
group	O
of	O
Lie	O
type	O
is	O
an	O
extension	O
of	O
a	O
group	O
of	O
"	O
diagonal	O
automorphisms	O
"	O
(	O
cyclic	O
except	O
for	O
,	O
when	O
it	O
has	O
order	O
4	O
)	O
,	O
a	O
group	O
of	O
"	O
field	O
automorphisms	O
"	O
(	O
always	O
cyclic	O
)	O
,	O
and	O
a	O
group	O
of	O
"	O
graph	O
automorphisms	O
"	O
(	O
of	O
order	O
1	O
or	O
2	O
except	O
for	O
,	O
when	O
it	O
is	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
3	O
points	O
)	O
.	O
</s>
<s>
These	O
extensions	O
are	O
not	O
always	O
semidirect	O
products	O
,	O
as	O
the	O
case	O
of	O
the	O
alternating	B-Algorithm
group	I-Algorithm
shows	O
;	O
a	O
precise	O
criterion	O
for	O
this	O
to	O
happen	O
was	O
given	O
in	O
2003	O
.	O
</s>
<s>
Group	O
Parameter	O
:	O
the	O
identity	O
and	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
;	O
one	O
corresponding	O
to	O
multiplication	O
by	O
an	O
invertible	O
element	O
in	O
the	O
ring	O
.	O
</s>
<s>
The	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
of	O
a	O
finite	O
simple	O
group	O
in	O
some	O
infinite	O
family	O
of	O
finite	O
simple	O
groups	O
can	O
almost	O
always	O
be	O
given	O
by	O
a	O
uniform	O
formula	O
that	O
works	O
for	O
all	O
elements	O
of	O
the	O
family	O
.	O
</s>
<s>
There	O
is	O
just	O
one	O
exception	O
to	O
this	O
:	O
the	O
alternating	B-Algorithm
group	I-Algorithm
has	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
of	O
order	O
4	O
,	O
rather	O
than	O
2	O
as	O
do	O
the	O
other	O
simple	O
alternating	B-Algorithm
groups	I-Algorithm
(	O
given	O
by	O
conjugation	O
by	O
an	O
odd	O
permutation	O
)	O
.	O
</s>
<s>
Equivalently	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
is	O
the	O
only	O
symmetric	B-Algorithm
group	I-Algorithm
with	O
a	O
non-trivial	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Then	O
any	O
two	O
Borel	O
subgroups	O
are	O
conjugate	O
by	O
an	O
inner	B-Algorithm
automorphism	I-Algorithm
,	O
so	O
to	O
study	O
outer	B-Algorithm
automorphisms	I-Algorithm
it	O
suffices	O
to	O
consider	O
automorphisms	O
that	O
fix	O
a	O
given	O
Borel	O
subgroup	O
.	O
</s>
<s>
Associated	O
to	O
the	O
Borel	O
subgroup	O
is	O
a	O
set	O
of	O
simple	O
roots	O
,	O
and	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
may	O
permute	O
them	O
,	O
while	O
preserving	O
the	O
structure	O
of	O
the	O
associated	O
Dynkin	O
diagram	O
.	O
</s>
<s>
In	O
this	O
way	O
one	O
may	O
identify	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
the	O
Dynkin	O
diagram	O
of	O
with	O
a	O
subgroup	O
of	O
.	O
</s>
<s>
has	O
a	O
very	O
symmetric	O
Dynkin	O
diagram	O
,	O
which	O
yields	O
a	O
large	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
of	O
,	O
namely	O
;	O
this	O
is	O
called	O
triality	O
.	O
</s>
<s>
The	O
term	O
outer	B-Algorithm
automorphism	I-Algorithm
lends	O
itself	O
to	O
word	O
play	O
:	O
the	O
term	O
outermorphism	O
is	O
sometimes	O
used	O
for	O
outer	B-Algorithm
automorphism	I-Algorithm
,	O
and	O
a	O
particular	O
geometry	O
on	O
which	O
acts	O
is	O
called	O
outer	O
space	O
.	O
</s>
