>>10117553what the other anons said, but normal is the "weakest" type of subgroup such that you can form a quotient and get a group. That is, a subgroup H of G is normal iff G/H is a group.
Obviously, this is a very useful criterion, since you can form a subgroup H consisting of "relations that you dont want in your group", find the smallest normal subgroup N containing H by taking the intersection of all normal subgroups containing H, then quotienting G by N will give you a group without those characteristics.
As an example in the abelian case: Consider a group G of order pn. Let g be any element, and consider its cyclic subgroup generated by g, that is {1,g,g^2,...}. Since G is finite, then there is some m with g^m=1, so it is indeed finite cyclic. But every finite cyclic group has elements of prime order q, where q divides m. If q=p for any of these, then there exists an element of order p, if it is not the case, then quotient G by {1,g,g^2,...} and you get a subgroup of order pn/m, but since p and m have no common factor, then you only take off primes "belonging" to n. So by quotienting you remove elements of order that isn't p. Hence you can inductively proof that there exists an element of order p by repeating this procedure.
