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Let $G$ be a group. Suppose $|G|= p^{2}q$, where $p$ and $q$ are distinct prime numbers satisfying $q ≢ 1 \mod p$. Which of the following is always true?

  1. $G$ has more than one $p$-Sylow subgroup.
  2. $G$ has a normal $p$-Sylow subgroup.
  3. The number of $q$-Sylow subgroups of $G$ is divisible by $p$.
  4. $G$ has a unique $q$-Sylow subgroup.
in Set Theory & Algebra by Boss (29.8k points) | 221 views

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