Exceptional Embedding of S5 in S6
In this article we shall show a peculiar property of .
We first note that there are several subgroups of isomorphic to
. In fact there is one for each
Let .
All of the above subgroups have two orbits in . But for
we show that there is a seventh embedding of
which acts transitively on
.
We now show that there is a transitive action of on X.
We look at the 5-Sylow subgroups of . They will be of order 5 each as
is of order 120 and 5 is the highest power of 5 dividing it. Hence, the only possibility for the 5-Sylow subgroups is to contain 5 cycles. Now, the number of r cycles of
is
. Hence there are 24 5-cycles. As each 5-Sylow subgroup consists of 4 cycles and as any two 5-Sylow subgroups has only identity in their intersection so number of 5-Sylow subgroups is
. Let the subgroups be
. The number of subgroups is compatible with the result of Sylow’s theorem which says that the number of 5-Sylow subgroups should be of the form
and should divide 120.
Let . We define an action of
on this set by conjugation. By Sylow’s theorem this is an action. Infact it is a transitive action. As
and
are bijective, we have established an action of
on
.
Let be the action homomorphism. If we show that
is injective then we see that there is an embedding of
in
which acts transitively on
.
Now we show that is injective.
Showing is injective is equivalent to showing that
is
. Now
is a normal subgroup of
. Hence
is either
,
or
. Here we use the fact that
is simple for all
. We know that
is isomorphic to
. As
so if
is either
or
then
but
as the action is transitive. Hence
is
.
It can be further showed that only n for which has a subgroup isomorphic to
which acts transitively on
is 6.
The above article is based on a lecture given at Chennai Mathematical Institute by Prof P. Vanchinathan.