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Math Help - Sylow p-group

  1. #1
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    Sylow p-group

    Could anyone help me on this question:
    Let G be finite group of order  |G| = p^a q^b ... t^d where p,q, ... t are distinct primes and a,b,.. d are positive integers. Suppose that G has only one Sylow p-group G_{p^a} for each prime p that divides |G|.
    Show that G = G_{p^a} \times G_{q^b} \times ... \times G_{t^d}

    Is it true that gcd(p^a, q^b)=1 , for any distinct primes?
    If it is true, can I use Sylow theorem and because G has only 1 Sylow p-group for each prime? Therefore
    G = G_{p^a} \times G_{q^b} \times ... \times G_{t^d}

    Thank you
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  2. #2
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    If H,K are normal subgroups then HK is a normal subgroup. Also if H\cap K = \{ e \} then HK \simeq H\times K.
    Thus, if \gcd (|H|,|K|) = 1 then H\cap K = \{ e \}. This result generalizes to finitely many groups.
    In your case you have the Sylow subgroups G_{p^a},G_{q^b}, ... all of them are normal because there is only one, and furthermore, all of them have trivial group intersection because their orders are relatively prime.
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