Thread: Subgroup with prime index

Prove that has no subgroup of prime index.

Originally Posted by didact273

Prove that has no subgroup of prime index.

you can prove a much stronger result: if and then this is a quick result of this well-known fact that for any group G and any subgroup H of G of finite index, there

exists a normal subgroup N of G contained in H such that

Originally Posted by NonCommAlg

you can prove a much stronger result: if and then this is a quick result of this well-known fact that for any group G and any subgroup H of G of finite index, there

exists a normal subgroup N of G contained in H such that

Can't I deduce from the fact that since is a simple group, the only possible subgroup is {(1)} and itself, there is no subgroup of index p?

Originally Posted by didact273

Can't I deduce from the fact that since is a simple group, the only possible subgroup is {(1)} and itself, there is no subgroup of index p?

the only possible (proper) normal subgroup is {(1)}. see the definition of a simple group again!