Prove that if A is a subset of B, then <A> is a subgroup of <B>. Give an example where A is a subest of B with A =/= B but <A> = <B>.
Please help.
$\displaystyle \left< A \right> = \bigcap_{A\subseteq H} H$ - the intersection of all subgroups of the group containing $\displaystyle A$.
$\displaystyle \left< B \right> = \bigcap_{B\subseteq H} H$ - the intersection of all subgroups of the group containing $\displaystyle B$.
Now since $\displaystyle A\subseteq B$ it follows,
$\displaystyle \left< A \right> = \bigcap_{A\subseteq H}H \subseteq \bigcap_{B\subseteq H} H = \left< B \right>$.
For the example, let $\displaystyle G = \mathbb{Z}_2$ then $\displaystyle \left< \{ [1] \} \right> = \left< G \right>$.