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Math Help - Sylow p-subgroups

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    Sylow p-subgroups

    Let G be a finite group and p be a prime.
    Prove that if Q \in Syl_p(G) and H is a subgroup of G containing Q then Q \in Syl_p(H) .
    Syl_p(G) is the set of Sylow p-subgroups of G.
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    Quote Originally Posted by dori1123 View Post
    Let G be a finite group and p be a prime.
    Prove that if Q \in Syl_p(G) and H is a subgroup of G containing Q then Q \in Syl_p(H) .
    Syl_p(G) is the set of Sylow p-subgroups of G.
    Write |G|=p^a\cdot n where p\not | n. Therefore |Q| = p^a. Since H\subseteq G it means |H| = p^b \cdot m where 0\leq b\leq a and m|n. But Q\subseteq H so |Q| divides |H|, thus, b=a. We see that Q is a Sylow subgroup because it has the largest exponent of p dividing |H|.
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