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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.

1 Answer

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Its mentioned that |G|=p^2q where p and q are two distinct prime numbers. By applying slyow’s 3rd theorem, 

we can see that

number of p-sylow subgroups= 1+2p   ….………………..(1)

number of q-sylow subgroups= 1+q ……………………….(2)

  1. clearly from (1) option A will always be correct
  2. its not always possible to have normal p-sylow subgroup as nothing about the elements is mentioned.
  3. from (2) is can be seen that option 3 is false
  4. again from (2) option D is false.
Answer:

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