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Thread: elements in a general product

  1. #1
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    elements in a general product

    Let $\displaystyle A,B $ be two subgroups of $\displaystyle G$ where only $\displaystyle A$ is normal.
    If $\displaystyle x \notin AB$, does it mean that $\displaystyle x \notin A$ and $\displaystyle x \notin B$?
    Suppose $\displaystyle x \notin AB$ and $\displaystyle y \notin A$. Does it implies $\displaystyle xy \notin A$?
    Last edited by deniselim17; Nov 9th 2017 at 02:55 AM.
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  2. #2
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    Re: elements in a general product

    First question

    $\displaystyle A\subseteq \text{AB}$

    therefore

    $\displaystyle \text {if } x \notin\text {AB }\text {then } x \notin A$

    Second question

    consider $\displaystyle A=B=<e>$ the trivial group
    Last edited by Idea; Nov 9th 2017 at 06:21 AM.
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  3. #3
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    Re: elements in a general product

    if $\displaystyle x \notin A, x \notin B$, does it mean that $\displaystyle x \notin AB$?
    A is normal subgroup of G.
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    Re: elements in a general product

    Quote Originally Posted by deniselim17 View Post
    if $\displaystyle x \notin A, x \notin B$, does it mean that $\displaystyle x \notin AB$?
    A is normal subgroup of G.
    No

    Definition: $\displaystyle \text{AB}=\{a b : a\in A, b\in B\} $ Correct?
    Last edited by Idea; Nov 13th 2017 at 03:20 AM.
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