Let be the quotient map and consider . (It is a subgroup of the simple group G/H.)
Prove that if is a normal subgroup of of prime index then for all subgroups of either
(i) is a subgroup of , or
(ii) and intersect
I tried to do this in 2 cases. The first case is to let be a subest of , then since is a subgroup of , it is a subgroup of . For case 2, I let not contained in , then try to show and intersect , but don't know how... please help, thank you.
The group has prime order, therefore, it only has no proper non-trivial subgroup. The natural projection defined by is a homomorphism. Thus, by the property of homomorphism, if is a subgroup of then is a subgroup of . This means that or . We have two cases, (i) (ii) . Now consider each case.