<s>
In	O
mathematics	O
,	O
in	O
the	O
realm	O
of	O
group	O
theory	O
,	O
a	O
class	B-Algorithm
automorphism	I-Algorithm
is	O
an	O
automorphism	O
of	O
a	O
group	O
that	O
sends	O
each	O
element	O
to	O
within	O
its	O
conjugacy	O
class	O
.	O
</s>
<s>
The	O
class	B-Algorithm
automorphisms	I-Algorithm
form	O
a	O
subgroup	O
of	O
the	O
automorphism	O
group	O
.	O
</s>
<s>
Every	O
inner	B-Algorithm
automorphism	I-Algorithm
is	O
a	O
class	B-Algorithm
automorphism	I-Algorithm
.	O
</s>
<s>
Every	O
class	B-Algorithm
automorphism	I-Algorithm
is	O
a	O
family	O
automorphism	O
and	O
a	O
quotientable	B-Algorithm
automorphism	I-Algorithm
.	O
</s>
<s>
Under	O
a	O
quotient	O
map	O
,	O
class	B-Algorithm
automorphisms	I-Algorithm
go	O
to	O
class	B-Algorithm
automorphisms	I-Algorithm
.	O
</s>
<s>
Every	O
class	B-Algorithm
automorphism	I-Algorithm
is	O
an	O
IA	B-Algorithm
automorphism	I-Algorithm
,	O
that	O
is	O
,	O
it	O
acts	O
as	O
identity	O
on	O
the	O
Abelianization	O
.	O
</s>
<s>
Every	O
class	B-Algorithm
automorphism	I-Algorithm
is	O
a	O
center-fixing	O
automorphism	O
,	O
that	O
is	O
,	O
it	O
fixes	O
all	O
points	O
in	O
the	O
center	O
.	O
</s>
<s>
Normal	O
subgroups	O
are	O
characterized	O
as	O
subgroups	O
invariant	O
under	O
class	B-Algorithm
automorphisms	I-Algorithm
.	O
</s>
<s>
For	O
infinite	O
groups	O
,	O
an	O
example	O
of	O
a	O
class	B-Algorithm
automorphism	I-Algorithm
that	O
is	O
not	O
inner	O
is	O
the	O
following	O
:	O
take	O
the	O
finitary	O
symmetric	O
group	O
on	O
countably	O
many	O
elements	O
and	O
consider	O
conjugation	O
by	O
an	O
infinitary	O
permutation	O
.	O
</s>
<s>
This	O
conjugation	O
defines	O
an	O
outer	B-Algorithm
automorphism	I-Algorithm
on	O
the	O
group	O
of	O
finitary	O
permutations	O
.	O
</s>
<s>
Finding	O
a	O
class	B-Algorithm
automorphism	I-Algorithm
in	O
the	O
stability	O
group	O
that	O
is	O
not	O
inner	O
boils	O
down	O
to	O
finding	O
a	O
cocycle	O
for	O
the	O
action	O
that	O
is	O
locally	O
a	O
coboundary	O
but	O
is	O
not	O
a	O
global	O
coboundary	O
.	O
</s>
