Question
Suppose $G$ is a finite group and $p$ is a prime that divides $|G| .$ Let $n$ denote the number of elements of $G$ that have order $p$. If the Sylow $p$ -subgroup of $G$ is normal, prove that $p$ divides $n+1$.
Step 1
Since \( P \) is normal in \( G \), it is invariant under conjugation by any element of \( G \). This means that all conjugates of any element in \( P \) remain in \( P \). Show more…
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