<s>
In	O
abstract	O
algebra	O
an	O
inner	B-Algorithm
automorphism	I-Algorithm
is	O
an	O
automorphism	O
of	O
a	O
group	O
,	O
ring	O
,	O
or	O
algebra	O
given	O
by	O
the	O
conjugation	O
action	O
of	O
a	O
fixed	O
element	O
,	O
called	O
the	O
conjugating	O
element	O
.	O
</s>
<s>
These	O
inner	B-Algorithm
automorphisms	I-Algorithm
form	O
a	O
subgroup	O
of	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
,	O
and	O
the	O
quotient	O
of	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
by	O
this	O
subgroup	O
is	O
defined	O
as	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
.	O
</s>
<s>
where	O
the	O
second	O
equality	O
is	O
given	O
by	O
the	O
insertion	O
of	O
the	O
identity	O
between	O
and	O
Furthermore	O
,	O
it	O
has	O
a	O
left	O
and	O
right	O
inverse	O
,	O
namely	O
Thus	O
,	O
is	O
bijective	B-Algorithm
,	O
and	O
so	O
an	O
isomorphism	O
of	O
with	O
itself	O
,	O
i.e.	O
</s>
<s>
An	O
inner	B-Algorithm
automorphism	I-Algorithm
is	O
any	O
automorphism	O
that	O
arises	O
from	O
conjugation	O
.	O
</s>
<s>
When	O
discussing	O
right	O
conjugation	O
,	O
the	O
expression	O
is	O
often	O
denoted	O
exponentially	O
by	O
This	O
notation	O
is	O
used	O
because	O
composition	B-Application
of	O
conjugations	O
satisfies	O
the	O
identity	O
:	O
for	O
all	O
This	O
shows	O
that	O
right	O
conjugation	O
gives	O
a	O
right	O
action	O
of	O
on	O
itself	O
.	O
</s>
<s>
The	O
composition	B-Application
of	O
two	O
inner	B-Algorithm
automorphisms	I-Algorithm
is	O
again	O
an	O
inner	B-Algorithm
automorphism	I-Algorithm
,	O
and	O
with	O
this	O
operation	O
,	O
the	O
collection	O
of	O
all	O
inner	B-Algorithm
automorphisms	I-Algorithm
of	O
is	O
a	O
group	O
,	O
the	O
inner	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
of	O
denoted	O
.	O
</s>
<s>
is	O
a	O
normal	O
subgroup	O
of	O
the	O
full	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
.	O
</s>
<s>
The	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
measures	O
,	O
in	O
a	O
sense	O
,	O
how	O
many	O
automorphisms	O
of	O
are	O
not	O
inner	O
.	O
</s>
<s>
Therefore	O
the	O
existence	O
and	O
number	O
of	O
inner	B-Algorithm
automorphisms	I-Algorithm
that	O
are	O
not	O
the	O
identity	O
mapping	O
is	O
a	O
kind	O
of	O
measure	O
of	O
the	O
failure	O
of	O
the	O
commutative	O
law	O
in	O
the	O
group	O
(	O
or	O
ring	O
)	O
.	O
</s>
<s>
By	O
associating	O
the	O
element	O
with	O
the	O
inner	B-Algorithm
automorphism	I-Algorithm
in	O
as	O
above	O
,	O
one	O
obtains	O
an	O
isomorphism	O
between	O
the	O
quotient	O
group	O
(	O
where	O
is	O
the	O
center	O
of	O
)	O
and	O
the	O
inner	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
:	O
</s>
<s>
This	O
is	O
a	O
consequence	O
of	O
the	O
first	O
isomorphism	O
theorem	O
,	O
because	O
is	O
precisely	O
the	O
set	O
of	O
those	O
elements	O
of	O
that	O
give	O
the	O
identity	O
mapping	O
as	O
corresponding	O
inner	B-Algorithm
automorphism	I-Algorithm
(	O
conjugation	O
changes	O
nothing	O
)	O
.	O
</s>
<s>
The	O
inner	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
of	O
a	O
group	O
,	O
,	O
is	O
trivial	O
(	O
i.e.	O
,	O
consists	O
only	O
of	O
the	O
identity	O
element	O
)	O
if	O
and	O
only	O
if	O
is	O
abelian	O
.	O
</s>
<s>
At	O
the	O
opposite	O
end	O
of	O
the	O
spectrum	O
,	O
the	O
inner	B-Algorithm
automorphisms	I-Algorithm
may	O
exhaust	O
the	O
entire	O
automorphism	B-Algorithm
group	I-Algorithm
;	O
a	O
group	O
whose	O
automorphisms	O
are	O
all	O
inner	O
and	O
whose	O
center	O
is	O
trivial	O
is	O
called	O
complete	O
.	O
</s>
<s>
If	O
the	O
inner	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
of	O
a	O
perfect	O
group	O
is	O
simple	O
,	O
then	O
is	O
called	O
quasisimple	O
.	O
</s>
<s>
An	O
automorphism	O
of	O
a	O
Lie	O
algebra	O
is	O
called	O
an	O
inner	B-Algorithm
automorphism	I-Algorithm
if	O
it	O
is	O
of	O
the	O
form	O
,	O
where	O
is	O
the	O
adjoint	O
map	O
and	O
is	O
an	O
element	O
of	O
a	O
Lie	O
group	O
whose	O
Lie	O
algebra	O
is	O
.	O
</s>
<s>
The	O
notion	O
of	O
inner	B-Algorithm
automorphism	I-Algorithm
for	O
Lie	O
algebras	O
is	O
compatible	O
with	O
the	O
notion	O
for	O
groups	O
in	O
the	O
sense	O
that	O
an	O
inner	B-Algorithm
automorphism	I-Algorithm
of	O
a	O
Lie	O
group	O
induces	O
a	O
unique	O
inner	B-Algorithm
automorphism	I-Algorithm
of	O
the	O
corresponding	O
Lie	O
algebra	O
.	O
</s>
<s>
If	O
is	O
the	O
group	O
of	O
units	O
of	O
a	O
ring	O
,	O
,	O
then	O
an	O
inner	B-Algorithm
automorphism	I-Algorithm
on	O
can	O
be	O
extended	O
to	O
a	O
mapping	O
on	O
the	O
projective	O
line	O
over	O
by	O
the	O
group	O
of	O
units	O
of	O
the	O
matrix	O
ring	O
,	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
inner	B-Algorithm
automorphisms	I-Algorithm
of	O
the	O
classical	O
groups	O
can	O
be	O
extended	O
in	O
that	O
way	O
.	O
</s>
