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Thread: Abstract Algebra: Lagrange's Theorem

  1. #1
    is up to his old tricks again! Jhevon's Avatar
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    Abstract Algebra: Lagrange's Theorem

    Hi all,

    Another problem I am not sure whether I am missing something or not.

    Problem:

    A finite group $\displaystyle G$ has elements of orders $\displaystyle p$ and $\displaystyle q$, where $\displaystyle p$ and $\displaystyle q$ are distinct primes. What can you conclude about $\displaystyle |G|$?


    Solution:

    I said that $\displaystyle |G| = kpq$ for some $\displaystyle k \in \mathbb{Z}_{>0} $. As a consequence of Lagrange's Theorem (the orders of the elements must divide |G|)

    does that seem like the observation they wanted me to make?

    Thanks
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  2. #2
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    Consider $\displaystyle \mathbb{Z}_{pq}, \mathbb{Z}_{2pq}, \mathbb{Z}_{3pq}, ... $ all of these groups have elements of orders $\displaystyle p,q$. While their orders are $\displaystyle pq,2pq,3pq,...$. Thus, it does not seem as there is a way to improve $\displaystyle |G| = kpq$, what your wrote.
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