<s>
They	O
are	O
multiply	O
transitive	O
permutation	B-Algorithm
groups	I-Algorithm
on	O
11	O
,	O
12	O
,	O
22	O
,	O
23	O
or	O
24	O
objects	O
.	O
</s>
<s>
introduced	O
the	O
group	O
M12	O
as	O
part	O
of	O
an	O
investigation	O
of	O
multiply	O
transitive	O
permutation	B-Algorithm
groups	I-Algorithm
,	O
and	O
briefly	O
mentioned	O
(	O
on	O
page	O
274	O
)	O
the	O
group	O
M24	O
,	O
giving	O
its	O
order	O
.	O
</s>
<s>
In	O
he	O
gave	O
further	O
details	O
,	O
including	O
explicit	O
generating	O
sets	O
for	O
his	O
groups	O
,	O
but	O
it	O
was	O
not	O
easy	O
to	O
see	O
from	O
his	O
arguments	O
that	O
the	O
groups	O
generated	O
are	O
not	O
just	O
alternating	B-Algorithm
groups	I-Algorithm
,	O
and	O
for	O
several	O
years	O
the	O
existence	O
of	O
his	O
groups	O
was	O
controversial	O
.	O
</s>
<s>
finally	O
removed	O
the	O
doubts	O
about	O
the	O
existence	O
of	O
these	O
groups	O
,	O
by	O
constructing	O
them	O
as	O
successive	O
transitive	O
extensions	O
of	O
permutation	B-Algorithm
groups	I-Algorithm
,	O
as	O
well	O
as	O
automorphism	O
groups	O
of	O
Steiner	O
systems	O
.	O
</s>
<s>
Mathieu	O
was	O
interested	O
in	O
finding	O
multiply	O
transitive	O
permutation	B-Algorithm
groups	I-Algorithm
,	O
which	O
will	O
now	O
be	O
defined	O
.	O
</s>
<s>
For	O
a	O
natural	O
number	O
k	O
,	O
a	O
permutation	B-Algorithm
group	I-Algorithm
G	O
acting	O
on	O
n	O
points	O
is	O
k-transitive	O
if	O
,	O
given	O
two	O
sets	O
of	O
points	O
a1	O
,	O
...	O
ak	O
and	O
b1	O
,	O
...	O
bk	O
with	O
the	O
property	O
that	O
all	O
the	O
ai	O
are	O
distinct	O
and	O
all	O
the	O
bi	O
are	O
distinct	O
,	O
there	O
is	O
a	O
group	O
element	O
g	O
in	O
G	O
which	O
maps	O
ai	O
to	O
bi	O
for	O
each	O
i	O
between	O
1	O
and	O
k	O
.	O
Such	O
a	O
group	O
is	O
called	O
sharply	O
k-transitive	O
if	O
the	O
element	O
g	O
is	O
unique	O
(	O
i.e.	O
</s>
<s>
These	O
are	O
the	O
only	O
two	O
5-transitive	O
groups	O
that	O
are	O
neither	O
symmetric	B-Algorithm
groups	I-Algorithm
nor	O
alternating	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
The	O
only	O
4-transitive	O
groups	O
are	O
the	O
symmetric	B-Algorithm
groups	I-Algorithm
Sk	O
for	O
k	O
at	O
least	O
4	O
,	O
the	O
alternating	B-Algorithm
groups	I-Algorithm
Ak	O
for	O
k	O
at	O
least	O
6	O
,	O
and	O
the	O
Mathieu	O
groups	O
M24	O
,	O
M23	O
,	O
M12	O
,	O
and	O
M11	O
.	O
</s>
<s>
It	O
is	O
a	O
classical	O
result	O
of	O
Jordan	O
that	O
the	O
symmetric	B-Algorithm
and	O
alternating	B-Algorithm
groups	I-Algorithm
(	O
of	O
degree	O
k	O
and	O
k+2	O
respectively	O
)	O
,	O
and	O
M12	O
and	O
M11	O
are	O
the	O
only	O
sharply	O
k-transitive	O
permutation	B-Algorithm
groups	I-Algorithm
for	O
k	O
at	O
least4	O
.	O
</s>
<s>
Important	O
examples	O
of	O
multiply	B-Algorithm
transitive	I-Algorithm
groups	I-Algorithm
are	O
the	O
2-transitive	B-Algorithm
groups	I-Algorithm
and	O
the	O
Zassenhaus	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
The	O
Zassenhaus	B-Algorithm
groups	I-Algorithm
notably	O
include	O
the	O
projective	O
general	O
linear	O
group	O
of	O
a	O
projective	O
line	O
over	O
a	O
finite	O
field	O
,	O
PGL(2,Fq )	O
,	O
which	O
is	O
sharply	O
3-transitive	O
(	O
see	O
cross	O
ratio	O
)	O
on	O
elements	O
.	O
</s>
<s>
M10	O
,	O
the	O
stabilizer	O
of	O
two	O
points	O
,	O
is	O
not	O
sporadic	O
,	O
but	O
is	O
an	O
almost	O
simple	O
group	O
whose	O
commutator	O
subgroup	O
is	O
the	O
alternating	B-Algorithm
group	I-Algorithm
A6	O
.	O
</s>
<s>
The	O
group	O
M24	O
is	O
the	O
permutation	O
automorphism	O
group	O
of	O
the	O
extended	B-Error_Name
binary	I-Error_Name
Golay	I-Error_Name
code	I-Error_Name
W	O
,	O
i.e.	O
,	O
the	O
group	O
of	O
permutations	O
on	O
the	O
24	O
coordinates	O
that	O
map	O
W	O
to	O
itself	O
.	O
</s>
<s>
All	O
the	O
Mathieu	O
groups	O
can	O
be	O
constructed	O
as	O
groups	O
of	O
permutations	O
on	O
the	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
.	O
</s>
<s>
There	O
is	O
a	O
natural	O
connection	O
between	O
the	O
Mathieu	O
groups	O
and	O
the	O
larger	O
Conway	O
groups	O
,	O
because	O
the	O
Leech	O
lattice	O
was	O
constructed	O
on	O
the	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
and	O
in	O
fact	O
both	O
lie	O
in	O
spaces	O
of	O
dimension	O
24	O
.	O
</s>
