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
$\displaystyle H$ 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 $\displaystyle H$.