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Thread: elements of a direct product

  1. #1
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    elements of a direct product

    Let $A$ be a group with subgroup $H=P \times Q$, where $P$ is infinite and $Q$ is finite.
    Suppose there exists $M \lhd_{f} A$ such that $p \notin M, q \notin M$, where $p \in P, q \in Q$.
    Is it true that $pq \notin M$?
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  2. #2
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    Re: elements of a direct product

    Try an example. Say for example $P=\mathbb{Z},Q=\mathbb{Z}_2$. Each uses addition as the group operator. M is the group of even integers (cross zero). $p=q=1$. Neither is in M. Their sum is in M.
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  3. #3
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    Re: elements of a direct product

    If I change $P$ and $Q$, to $P=\langle p \rangle$, an infinite cyclic subgroup and $Q$ is finite central subgroup in $A$.
    It is clear that $P \cap Q=1$.
    can $pq \notin M$?
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  4. #4
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    Re: elements of a direct product

    Quote Originally Posted by deniselim17 View Post
    If I change $P$ and $Q$, to $P=\langle p \rangle$, an infinite cyclic subgroup and $Q$ is finite central subgroup in $A$.
    It is clear that $P \cap Q=1$.
    can $pq \notin M$?
    I am a bit rusty with my algebra. I think I misunderstood the question. I will leave it to someone else to answer from here.
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