08 Jul 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
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.