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\(\def\Z{\mathbb{Z}} \def\zn{\mathbb{Z}_n} \def\znc{\mathbb{Z}_n^\times} \def\R{\mathbb{R}} \def\Q{\mathbb{Q}} \def\C{\mathbb{C}} \def\N{\mathbb{N}} \def\M{\mathbb{M}} \def\G{\mathcal{G}} \def\0{\mathbf 0} \def\Gdot{\langle G, \cdot\,\rangle} \def\phibar{\overline{\phi}} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Ker}{Ker} \def\siml{\sim_L} \def\simr{\sim_R} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section8.4Exercises

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.

  1. \(|2\Z/6\Z|\)
  2. \(|H|\text{,}\) for \(H=2+\langle 6\rangle \subseteq \Z_{12}\)
  3. \(o(2+\langle 6\rangle)\) in \(\Z_{12}/\langle 6\rangle\)
  4. \(\langle fH\rangle \) in \(D_4/H\text{,}\) where \(H=\{e,r^2\}\)
  5. \(|(\Z_6\times \Z_8)/(\langle 3\rangle\times \langle 2\rangle)|\)
  6. \(|(\Z_{15} \times \Z_{24})/\langle (5,4)\rangle|\)
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.)

  1. \(\Z/14\Z\)
  2. \(3\Z/12\Z\)
  3. \(S_8/A_8\)
  4. \((\Z_4 \times \Z_{15})/(\langle 2 \rangle \times \langle 3 \rangle )\)
  5. \(D_4/\langle r^2 \rangle\)
5

Let \(H\unlhd G\) with index \(k\text{,}\) and let \(a\in G\text{.}\) Prove that \(a^k\in H\text{.}\)