abstract algebra

**dori1123** · October 4th 2008, 03:01 PM

Prove that if A is a subset of B, then <A> is a subgroup of . Give an example where A is a subest of B with A =/= B but <A> = .

Please help.

**ThePerfectHacker** · October 4th 2008, 03:59 PM

Originally Posted by dori1123

Prove that if A is a subset of B, then <A> is a subgroup of . Give an example where A is a subest of B with A =/= B but <A> = .

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

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

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

**dori1123** · October 4th 2008, 05:42 PM

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?

**ThePerfectHacker** · October 4th 2008, 06:00 PM

Originally Posted by dori1123

Is $\mathbb{Z}_2$ the same as Z/2Z?

Yes

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