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Math Help - p-sylow subgroup

  1. #1
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    p-sylow subgroup

    Let o(G)=pq, where p>q are primes.
    Since p divides o(G), G contains an element b of order p.
    Let S=<b>.
    Since S is a p-sylow subgroup of G, the number of conjugates is 1+kp for some k \geqslant 0.
    But  1+kp=[G:N(S)] which divides o(G)=pq.
    Since (1+kp,p)=1, then 1+kp divides q.
    Since q<p, then k=0 and S is normal in G.

    I don't understand why S is normal in G. If it is because [G:N(S)]=1, why?
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  2. #2
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    Sylow's theorem explains that. You know that p-sylow groups are all conjugate, so if there is only one p-sylow S in G, then it is normal.
    The reason is that for all x in G, xSx^(-1) is a p-sylow too (his order is p), so it is S.
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