If h is even, then ghg^-1 is an even element in S_n for every g. So A_n is normal in S_n.

If det(A) = 1, then for every invertible matrix B, the product BAB^-1 has determinant 1. Hence SL_n(R) is normal in GL_n(R).

The center of a group is a normal subgroup since all its elements commute with every element in the group.

Suppose that G is a commutative group and H is a subgroup. Then for every g G and h H, we have ghg^-1 = h, so H is a normal subgroup of G.

This shows that every subgroup of a commutative group is normal.