<s>
In	O
mathematics	O
and	O
group	O
theory	O
,	O
a	O
block	B-Algorithm
system	I-Algorithm
for	O
the	O
action	O
of	O
a	O
group	O
G	O
on	O
a	O
set	O
X	O
is	O
a	O
partition	O
of	O
X	O
that	O
is	O
G-invariant	O
.	O
</s>
<s>
The	O
action	O
of	O
G	O
on	O
X	O
induces	O
a	O
natural	O
action	O
of	O
G	O
on	O
any	O
block	B-Algorithm
system	I-Algorithm
for	O
X	O
.	O
</s>
<s>
The	O
set	O
of	O
orbits	O
of	O
the	O
G-set	O
X	O
is	O
an	O
example	O
of	O
a	O
block	B-Algorithm
system	I-Algorithm
.	O
</s>
<s>
The	O
corresponding	O
equivalence	O
relation	O
is	O
the	O
smallest	O
G-invariant	O
equivalence	O
on	O
X	O
such	O
that	O
the	O
induced	O
action	O
on	O
the	O
block	B-Algorithm
system	I-Algorithm
is	O
trivial	O
.	O
</s>
<s>
The	O
partition	O
into	O
singleton	O
sets	O
is	O
a	O
block	B-Algorithm
system	I-Algorithm
and	O
if	O
X	O
is	O
non-empty	O
then	O
the	O
partition	O
into	O
one	O
set	O
X	O
itself	O
is	O
a	O
block	B-Algorithm
system	I-Algorithm
as	O
well	O
(	O
if	O
X	O
is	O
a	O
singleton	O
set	O
then	O
these	O
two	O
partitions	O
are	O
identical	O
)	O
.	O
</s>
<s>
A	O
transitive	O
(	O
and	O
thus	O
non-empty	O
)	O
G-set	O
X	O
is	O
said	O
to	O
be	O
primitive	B-Algorithm
if	O
it	O
has	O
no	O
other	O
block	B-Algorithm
systems	I-Algorithm
.	O
</s>
<s>
Each	O
element	O
of	O
some	O
block	B-Algorithm
system	I-Algorithm
is	O
called	O
a	O
block	O
.	O
</s>
<s>
In	O
the	O
other	O
direction	O
,	O
if	O
the	O
set	O
B	O
satisfies	O
the	O
given	O
condition	O
then	O
the	O
system	O
{	O
gB	O
|	O
g	O
∈	O
G}	O
together	O
with	O
the	O
complement	O
of	O
the	O
union	O
of	O
these	O
sets	O
is	O
a	O
block	B-Algorithm
system	I-Algorithm
containing	O
B	O
.	O
</s>
<s>
In	O
particular	O
,	O
if	O
B	O
is	O
a	O
block	O
then	O
gB	O
is	O
a	O
block	O
for	O
any	O
g	O
∈	O
G	O
,	O
and	O
if	O
G	O
acts	O
transitively	O
on	O
X	O
then	O
the	O
set	O
{	O
gB	O
|	O
g	O
∈	O
G}	O
is	O
a	O
block	B-Algorithm
system	I-Algorithm
on	O
X	O
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
blocks	O
containing	O
x	O
and	O
contained	O
in	O
G.x	O
are	O
in	O
one-to-one	B-Algorithm
correspondence	I-Algorithm
with	O
the	O
subgroups	O
of	O
G	O
containing	O
Gx	O
.	O
</s>
<s>
In	O
particular	O
,	O
if	O
the	O
G-set	O
X	O
is	O
transitive	O
then	O
the	O
blocks	O
containing	O
x	O
are	O
in	O
one-to-one	B-Algorithm
correspondence	I-Algorithm
with	O
the	O
subgroups	O
of	O
G	O
containing	O
Gx	O
.	O
</s>
<s>
In	O
this	O
case	O
the	O
G-set	O
X	O
is	O
primitive	B-Algorithm
if	O
and	O
only	O
if	O
either	O
the	O
group	O
action	O
is	O
trivial	O
(	O
then	O
X	O
=	O
 { x } 	O
)	O
or	O
the	O
stabilizer	O
Gx	O
is	O
a	O
maximal	O
subgroup	O
of	O
G	O
(	O
then	O
the	O
stabilizers	O
of	O
all	O
elements	O
of	O
X	O
are	O
the	O
maximal	O
subgroups	O
of	O
G	O
conjugate	O
to	O
Gx	O
because	O
Ggx	O
=	O
g	O
⋅	O
Gx	O
⋅	O
g−1	O
)	O
.	O
</s>
