Suppose that G is a group and H is a subgroup of G. Define: H is a normal subgroup of G.?

Answer 1

Again this should be in your book. There should be a theorem with 4 different ways of saying this that are equivalent.

N is normal in G if gN=Ng for all g in G.

Answer 2

There are lots of equivalent definitions for this.

The most common one is: H is a subgroup of G such that xH = Hx for all x in G.

An equivalent definition: H is a subgroup of G such that xHx^(-1) is a subset of H for all x in G.

Another equivalent definition: H is a subgroup of G such that the normalizer of H in G is equal to H itself.

Another equivalent definition: H is a subgroup of G that is a union of conjugacy classes in G.