<s>
In	O
mathematics	O
,	O
a	O
permutation	B-Algorithm
group	I-Algorithm
G	O
acting	O
on	O
a	O
non-empty	O
finite	O
set	O
X	O
is	O
called	O
primitive	O
if	O
G	O
acts	O
transitively	O
on	O
X	O
and	O
the	O
only	O
partitions	O
the	O
G-action	O
preserves	O
are	O
the	O
trivial	O
partitions	O
into	O
either	O
a	O
single	O
set	O
or	O
into	O
|X|	O
singleton	O
sets	O
.	O
</s>
<s>
Otherwise	O
,	O
if	O
G	O
is	O
transitive	O
and	O
G	O
does	O
preserve	O
a	O
nontrivial	O
partition	O
,	O
G	O
is	O
called	O
imprimitive	B-Algorithm
.	O
</s>
<s>
While	O
primitive	B-Algorithm
permutation	I-Algorithm
groups	I-Algorithm
are	O
transitive	O
,	O
not	O
all	O
transitive	O
permutation	B-Algorithm
groups	I-Algorithm
are	O
primitive	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
if	O
a	O
permutation	B-Algorithm
group	I-Algorithm
preserves	O
only	O
trivial	O
partitions	O
,	O
it	O
is	O
transitive	O
,	O
except	O
in	O
the	O
case	O
of	O
the	O
trivial	O
group	O
acting	O
on	O
a	O
2-element	O
set	O
.	O
</s>
<s>
It	O
follows	O
that	O
,	O
if	O
is	O
a	O
prime	O
number	O
greater	O
than	O
3	O
,	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
and	O
the	O
alternating	B-Algorithm
group	I-Algorithm
of	O
degree	O
are	O
not	O
solvable	O
,	O
since	O
their	O
order	O
are	O
greater	O
than	O
Abel	O
–	O
Ruffini	O
theorem	O
results	O
from	O
this	O
and	O
the	O
fact	O
that	O
there	O
are	O
polynomials	O
with	O
a	O
symmetric	B-Algorithm
Galois	O
group	O
.	O
</s>
<s>
An	O
equivalent	O
definition	O
of	O
primitivity	O
relies	O
on	O
the	O
fact	O
that	O
every	O
transitive	O
action	O
of	O
a	O
group	O
G	O
is	O
isomorphic	O
to	O
an	O
action	O
arising	O
from	O
the	O
canonical	O
action	O
of	O
G	O
on	O
the	O
set	O
G/H	O
of	O
cosets	O
for	O
H	O
a	O
subgroup	O
of	O
G	O
.	O
A	O
group	O
action	O
is	O
primitive	O
if	O
it	O
is	O
isomorphic	O
to	O
G/H	O
for	O
a	O
maximal	O
subgroup	O
H	O
of	O
G	O
,	O
and	O
imprimitive	B-Algorithm
otherwise	O
(	O
that	O
is	O
,	O
if	O
there	O
is	O
a	O
proper	O
subgroup	O
K	O
of	O
G	O
of	O
which	O
H	O
is	O
a	O
proper	O
subgroup	O
)	O
.	O
</s>
<s>
These	O
imprimitive	B-Algorithm
actions	O
are	O
examples	O
of	O
induced	O
representations	O
.	O
</s>
<s>
The	O
numbers	O
of	O
primitive	B-Algorithm
groups	I-Algorithm
of	O
small	O
degree	O
were	O
stated	O
by	O
Robert	O
Carmichael	O
in	O
1937	O
:	O
</s>
<s>
There	O
are	O
a	O
large	O
number	O
of	O
primitive	B-Algorithm
groups	I-Algorithm
of	O
degree	O
16	O
.	O
</s>
<s>
As	O
Carmichael	O
notes	O
,	O
all	O
of	O
these	O
groups	O
,	O
except	O
for	O
the	O
symmetric	B-Algorithm
and	O
alternating	B-Algorithm
group	I-Algorithm
,	O
are	O
subgroups	O
of	O
the	O
affine	O
group	O
on	O
the	O
4-dimensional	O
space	O
over	O
the	O
2-element	O
finite	O
field	O
.	O
</s>
<s>
The	O
symmetric	B-Algorithm
group	I-Algorithm
acting	O
on	O
the	O
set	O
is	O
primitive	O
for	O
every	O
n	O
and	O
the	O
alternating	B-Algorithm
group	I-Algorithm
acting	O
on	O
the	O
set	O
is	O
primitive	O
for	O
everyn>2	O
.	O
</s>
