1. ## abstract algebra

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>.

2. Originally Posted by dori1123
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>.
$\left< A \right> = \bigcap_{A\subseteq H} H$ - the intersection of all subgroups of the group containing $A$.
$\left< B \right> = \bigcap_{B\subseteq H} H$ - the intersection of all subgroups of the group containing $B$.

Now since $A\subseteq B$ it follows,
$\left< A \right> = \bigcap_{A\subseteq H}H \subseteq \bigcap_{B\subseteq H} H = \left< B \right>$.

For the example, let $G = \mathbb{Z}_2$ then $\left< \{ [1] \} \right> = \left< G \right>$.

3. Originally Posted by ThePerfectHacker
For the example, let $G = \mathbb{Z}_2$ then $\left< \{ [1] \} \right> = \left< G \right>$.
Is $\mathbb{Z}_2$ the same as Z/2Z?

4. Originally Posted by dori1123
Is $\mathbb{Z}_2$ the same as Z/2Z?
Yes