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Thread: subset

  1. #1
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    subset

    may i know what is the different between the subgroup <H> of G and the subset H of G.

    i dont really get what it means for
    the subgroup <H> conincides with the intersection of all subgroups of G which contain the subset H
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  2. #2
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    $\displaystyle H$ is a subset of $\displaystyle G$ if every element of $\displaystyle H$ is an element of $\displaystyle G$.

    $\displaystyle H$ is a subgroup of $\displaystyle G$ if $\displaystyle H$ is a subset of $\displaystyle G$ and $\displaystyle H$ is a group.

    The subgroup generated by $\displaystyle H$ is the smallest subgroup of $\displaystyle G$ containing $\displaystyle H$. Informally, you need to throw as little "stuff" as possible into $\displaystyle H$ until you get a group. More formally, take the intersection of all subgroups of $\displaystyle G$ containing $\displaystyle H$. Since arbitrary intersections of a group form a group, you get the smallest subgroup of $\displaystyle G$ that contains $\displaystyle H$.
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