Thread: Abstract help

Need some guidance on some questions.

1. Let G be a finite group and let p be a prime. If the order of G <p^2, show that any subgroup of order p is normal in G.

I am unsure of how to do this problem, any help would be great.

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.

I know there are some subgroup tests but I am not sure which one is the easiest to use in this situation.

b. Determine the order of G/H.

Wouldn't the order of G/H be infinite?

Thank you in advance. I am terrible with proofs.

Originally Posted by jonnyfive

1. Let G be a finite group and let p be a prime. If the order of G <p^2, show that any subgroup of order p is normal in G.

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 .

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.
.

Here and .
Therefore, .