<s>
In	O
mathematics	O
,	O
the	O
O'Nan–Scott	B-Algorithm
theorem	I-Algorithm
is	O
one	O
of	O
the	O
most	O
influential	O
theorems	O
of	O
permutation	B-Algorithm
group	I-Algorithm
theory	O
;	O
the	O
classification	O
of	O
finite	O
simple	O
groups	O
is	O
what	O
makes	O
it	O
so	O
useful	O
.	O
</s>
<s>
Originally	O
the	O
theorem	O
was	O
about	O
maximal	O
subgroups	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
theorem	O
states	O
that	O
a	O
maximal	O
subgroup	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
Sym(Ω )	O
,	O
where	O
|Ω|	O
=	O
n	O
,	O
is	O
one	O
of	O
the	O
following	O
:	O
</s>
<s>
primitive	B-Algorithm
(	O
that	O
is	O
,	O
preserves	O
no	O
nontrivial	O
partition	O
)	O
and	O
of	O
one	O
of	O
the	O
following	O
types	O
:	O
</s>
<s>
Peter	O
J	O
.	O
Cameron	O
seems	O
to	O
have	O
been	O
the	O
first	O
to	O
recognize	O
that	O
the	O
real	O
power	O
in	O
the	O
O'Nan–Scott	B-Algorithm
theorem	I-Algorithm
is	O
in	O
the	O
ability	O
to	O
split	O
the	O
finite	O
primitive	B-Algorithm
groups	I-Algorithm
into	O
various	O
types	O
.	O
</s>
<s>
The	O
theorem	O
is	O
now	O
a	O
standard	O
part	O
of	O
textbooks	O
on	O
permutation	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
HA	O
(	O
holomorph	O
of	O
an	O
abelian	O
group	O
)	O
:	O
These	O
are	O
the	O
primitive	B-Algorithm
groups	I-Algorithm
which	O
are	O
subgroups	O
of	O
the	O
affine	O
general	O
linear	O
group	O
AGL(d,p )	O
,	O
for	O
some	O
prime	O
p	O
and	O
positive	O
integer	O
d	O
≥	O
1	O
.	O
</s>
<s>
For	O
such	O
a	O
group	O
G	O
to	O
be	O
primitive	B-Algorithm
,	O
it	O
must	O
contain	O
the	O
subgroup	O
of	O
all	O
translations	O
,	O
and	O
the	O
stabilizer	O
G0	O
in	O
G	O
of	O
the	O
zero	O
vector	O
must	O
be	O
an	O
irreducible	O
subgroup	O
of	O
GL(d,p )	O
.	O
</s>
<s>
Primitive	B-Algorithm
groups	I-Algorithm
of	O
type	O
HA	O
are	O
characterized	O
by	O
having	O
a	O
unique	O
minimal	O
normal	O
subgroup	O
which	O
is	O
elementary	O
abelian	O
and	O
acts	O
regularly	O
.	O
</s>
<s>
The	O
action	O
of	O
M	O
is	O
primitive	B-Algorithm
and	O
if	O
we	O
take	O
α	O
=	O
1T	O
we	O
have	O
Mα	O
=	O
{( 	O
t	O
,	O
t	O
)	O
|t	O
∈	O
T}	O
,	O
which	O
includes	O
Inn(T )	O
on	O
Ω	O
.	O
</s>
<s>
A	O
primitive	B-Algorithm
group	I-Algorithm
of	O
type	O
HS	O
is	O
then	O
any	O
group	O
G	O
such	O
that	O
M	O
≅	O
T.Inn(T )	O
≤	O
G	O
≤	O
T.Aut(T )	O
.	O
</s>
<s>
A	O
primitive	B-Algorithm
group	I-Algorithm
of	O
type	O
HC	O
is	O
a	O
group	O
G	O
such	O
that	O
M	O
≤	O
G	O
≤	O
Tk.Aut(Tk )	O
and	O
G	O
induces	O
a	O
subgroup	O
of	O
Aut(Tk )	O
=	O
Aut(T )	O
wrSk	O
which	O
acts	O
transitively	O
on	O
the	O
set	O
of	O
k	O
simple	O
direct	O
factors	O
of	O
Tk	O
.	O
</s>
<s>
A	O
group	O
of	O
type	O
HC	O
preserves	O
a	O
product	O
structure	O
Ω	O
=	O
Δk	O
where	O
Δ	O
=	O
T	O
and	O
G≤	O
HwrSk	O
where	O
H	O
is	O
a	O
primitive	B-Algorithm
group	I-Algorithm
of	O
type	O
HS	O
on	O
Δ	O
.	O
</s>
<s>
The	O
conditions	O
required	O
to	O
get	O
primitivity	O
imply	O
that	O
k≥	O
6	O
so	O
the	O
smallest	O
degree	O
of	O
such	O
a	O
primitive	B-Algorithm
group	I-Algorithm
is	O
606	O
.	O
</s>
<s>
We	O
are	O
not	O
told	O
anything	O
about	O
what	O
the	O
action	O
is	O
,	O
other	O
than	O
that	O
it	O
is	O
primitive	B-Algorithm
.	O
</s>
<s>
Analysis	O
of	O
this	O
type	O
requires	O
knowing	O
about	O
the	O
possible	O
primitive	B-Algorithm
actions	O
of	O
almost	O
simple	O
groups	O
,	O
which	O
is	O
equivalent	O
to	O
knowing	O
the	O
maximal	O
subgroups	O
of	O
almost	O
simple	O
groups	O
.	O
</s>
<s>
A	O
primitive	B-Algorithm
group	I-Algorithm
of	O
type	O
SD	O
is	O
a	O
group	O
G	O
≤	O
W	O
such	O
that	O
N	O
◅	O
G	O
and	O
G	O
induces	O
a	O
primitive	B-Algorithm
subgroup	O
of	O
Sk	O
on	O
the	O
k	O
simple	O
direct	O
factors	O
of	O
N	O
.	O
</s>
<s>
CD	O
(	O
compound	O
diagonal	O
)	O
:	O
Here	O
Ω	O
=	O
Δk	O
and	O
G	O
≤	O
HwrSk	O
where	O
H	O
is	O
a	O
primitive	B-Algorithm
group	I-Algorithm
of	O
type	O
SD	O
on	O
Δ	O
with	O
minimal	O
normal	O
subgroup	O
Tl	O
.	O
</s>
<s>
PA	O
(	O
product	O
action	O
)	O
:	O
Here	O
Ω	O
=	O
Δk	O
and	O
G	O
≤	O
HwrSk	O
where	O
H	O
is	O
a	O
primitive	B-Algorithm
almost	O
simple	O
group	O
on	O
with	O
socle	O
T	O
.	O
Thus	O
G	O
has	O
a	O
product	O
action	O
on	O
Ω	O
.	O
</s>
