1. ## 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.

2. 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 $\displaystyle G$ has a subgroup of order $\displaystyle p$ it means $\displaystyle |G|$ is divisible by $\displaystyle p$. But since $\displaystyle |G|<p^2$ it means $\displaystyle |G| = pk$ where $\displaystyle 1\leq k < p$. Let $\displaystyle n$ be the number of Sylow $\displaystyle p$-subgroups. Then we know that $\displaystyle n|k$ and $\displaystyle n\equiv 1(\bmod p)$. This forces that $\displaystyle n=1$. Therefore, there is only one Sylow $\displaystyle p$-subgroup which means that the unique subgroup of order $\displaystyle p$ is normal in $\displaystyle G$.

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 $\displaystyle G = \mathbb{Z}\times \mathbb{Z}$ and $\displaystyle H = 2\mathbb{Z} \times 2\mathbb{Z}$.
Therefore, $\displaystyle G/H = (\mathbb{Z}\times \mathbb{Z})/ (2\mathbb{Z} \times 2\mathbb{Z}) \simeq (\mathbb{Z}/2\mathbb{Z}) \times (\mathbb{Z}/2\mathbb{Z}) = \mathbb{Z}_2 \times \mathbb{Z}_2$.
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