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Thread: Sylow Subgroups

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    Sylow Subgroups

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