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Math Help - ı need urgent help

  1. #1
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    Exclamation ı need urgent help

    1. Let p be prime, and G be a finite group. If every element of G has order a power of p, then |G| = p^n for some n≥0. (Hint: Use Cauchy’s theorem.)


    2. Tell as much as possible about the subgroups of a group of order 30 and of a group of order 40.


    ( please help me... its emergency for me )
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  2. #2
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    Quote Originally Posted by bogazichili View Post
    1. Let p be prime, and G be a finite group. If every element of G has order a power of p, then |G| = p^n for some n≥0. (Hint: Use Cauchy’s theorem.)
    Assume that there is q\not = p such that q divides order of |G|. Then by Cauchy's theorem there exists and element of order q which is not a power of p - a contradiction. Thus, all prime divisors of |G| must be equal to p and so |G| = p^n.

    2. Tell as much as possible about the subgroups of a group of order 30 and of a group of order 40.
    I am not sure exactly what you are looking for.
    But for order 30 it can be shown (I looked up a classification table) that these groups are: \mathbb{Z}_{30}, D_{15}, D_5 \times \mathbb{Z}_3, D_3 \times \mathbb{Z}_5.
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