subset

• Feb 4th 2011, 08:11 AM
alexandrabel90
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
• Feb 4th 2011, 01:14 PM
DrSteve
\$\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\$.