Since has a subgroup of order it means is divisible by . But since it means where . Let be the number of Sylow -subgroups. Then we know that and . This forces that . Therefore, there is only one Sylow -subgroup which means that the unique subgroup of order is normal in .

Here and .2. Let G= Z+Z(External direct product of integers) and H= {(x,y) such that x and y are even integers}.

a. Show that H is a subgroup of G.

b. Determine the order of G/H.

.

Therefore, .