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Math Help - Abstract help

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by jonnyfive View Post
    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 G has a subgroup of order p it means |G| is divisible by p. But since |G|<p^2 it means |G| = pk where 1\leq k < p. Let n be the number of Sylow p-subgroups. Then we know that n|k and n\equiv 1(\bmod p). This forces that n=1. Therefore, there is only one Sylow p-subgroup which means that the unique subgroup of order p is normal in 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 G = \mathbb{Z}\times \mathbb{Z} and H = 2\mathbb{Z} \times 2\mathbb{Z}.
    Therefore, 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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