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Math Help - Sylow Subgroups

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    Sylow Subgroups

    Let G be a finite group. Let n_{p} denote the number of p-sylow subgroups. Prove that if  n_{p} \ncong 1 \ (mod \ p^{2} )       then there exists p-sylow subgroups P and Q of G such that  |P:P \cap Q|=|Q:Q \cap P|=p
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    Quote Originally Posted by Chandru1 View Post
    Let G be a finite group. Let n_{p} denote the number of p-sylow subgroups. Prove that if  n_{p} \ncong 1 \ (mod \ p^{2} ) then there exists p-sylow subgroups P and Q of G such that  |P:P \cap Q|=|Q:Q \cap P|=p
    if you take a look at the proof of Sylow theorems, you'll see that it is proved that if Q is a p-subgroup (Sylow or non-Sylow) and if P_1, \cdots , P_k, \ \ k=n_p, are the p-Sylow subgroups of G, then
    \sum_{i=1}^s |Q : P_i \cap Q| = k, for some s \leq k. the result now follows easily from the given condition k \ncong 1 \mod p^2.
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