8.4: Exercises
1. Let \(G\) be a group and let \(H\leq G\) have index 2. Prove that \(H\unlhd G\text{.}\)
2. Let \(G\) be an abelian group with \(N\unlhd G\text{.}\) Prove that \(G/N\) is abelian .
3. Find the following.
- \(|2\mathbb{Z}/6\mathbb{Z}|\)
- \(|H|\text{,}\) for \(H=2+\langle 6\rangle \subseteq \mathbb{Z}_{12}\)
- \(o(2+\langle 6\rangle)\) in \(\mathbb{Z}_{12}/\langle 6\rangle\)
- \(\langle fH\rangle \) in \(D_4/H\text{,}\) where \(H=\{e,r^2\}\)
- \(|(\mathbb{Z}_6\times \mathbb{Z}_8)/(\langle 3\rangle\times \langle 2\rangle)|\)
- \(|(\mathbb{Z}_{15} \times \mathbb{Z}_{24})/\langle (5,4)\rangle|\)
4. For each of the following, find a familiar group to which the given group is isomorphic. (Hint: Consider the group order, properties such as abelianness and cyclicity, group tables, orders of elements, etc.)
- \(\mathbb{Z}/14\mathbb{Z}\)
- \(3\mathbb{Z}/12\mathbb{Z}\)
- \(S_8/A_8\)
- \((\mathbb{Z}_4 \times \mathbb{Z}_{15})/(\langle 2 \rangle \times \langle 3 \rangle )\)
- \(D_4/\langle r^2 \rangle\)
5. Let \(H\unlhd G\) with index \(k\text{,}\) and let \(a\in G\text{.}\) Prove that \(a^k\in H\text{.}\)