7.4: Exercises

1. How many distinct partitions of the set \(S=\{a,b,c,d\}\) are there? You do not need to list them. (Yes, you can find this answer online. But I recommend doing the work yourself for practice working with partitions!)

2. 

1. Let \(n\in \mathbb{Z}^+\text{.}\) Prove that \(\equiv_n\) is an equivalence relation on \(\mathbb{Z}\text{.}\)
2. The cells of the induced partition of \(\mathbb{Z}\) are called the residue classes (or congruence classes) of \(\mathbb{Z}\) modulo \(n\). Using set notation of the form \(\{\ldots,\#, \#,\#,\ldots\}\) for each class, write down the residue classes of \(\mathbb{Z}\) modulo \(4\text{.}\)

3. Let \(G\) be a group with subgroup \(H\text{.}\) Prove that \(\sim_R\) is an equivalence relation on \(G\text{.}\)

4. Find the indices of:

1. \(H=\langle (15)(24)\rangle\) in \(S_5\)
2. \(K=\langle (2354)(34)\rangle\) in \(S_6\)
3. \(A_n\) in \(S_n\)

5. For each subgroup \(H\) of group \(G\text{,}\) (i) find the left and the right cosets of \(H\) in \(G\text{,}\) (ii) decide whether or not \(H\) is normal in \(G\text{,}\) and (iii) find \((G:H)\text{.}\)

Write all permutations using disjoint cycle notation, and write all dihedral group elements using standard form.

1. \(H=6\mathbb{Z}\) in \(G=2\mathbb{Z}\)
2. \(H=\langle 4\rangle\) in \(\mathbb{Z}_{20}\)
3. \(H=\langle (23)\rangle\) in \(G=S_3\)
4. \(H=\langle r\rangle\) in \(G=D_4\)
5. \(H=\langle f\rangle\) in \(G=D_4\)

6. For each of the following, give an example of a group \(G\) with a subgroup \(H\) that matches the given conditions. If no such example exists, prove that.

1. A group \(G\) with subgroup \(H\) such that \(|G/H|=1\text{.}\)
2. A finite group \(G\) with subgroup \(H\) such that \(|G/H|=|G|\text{.}\)
3. An abelian group \(G\) of order \(8\) containing a non-normal subgroup \(H\) of order 2.
4. A group \(G\) of order 8 containing a normal subgroup of order \(2\text{.}\)
5. A nonabelian group \(G\) of order 8 containing a normal subgroup of index \(2\text{.}\)
6. A group \(G\) of order 8 containing a subgroup of order \(3\text{.}\)
7. An infinite group \(G\) containing a subgroup \(H\) of finite index.
8. An infinite group \(G\) containing a finite nontrivial subgroup \(H\text{.}\)

7. True/False. For each of the following, write T if the statement is true; otherwise, write F. You do NOT need to provide explanations or show work for this problem. Throughout, let \(G\) be a group with subgroup \(H\) and elements \(a,b\in G\text{.}\)

1. If \(a\in bH\) then \(aH\) must equal \(bH\text{.}\)
2. \(aH\) must equal \(Ha\text{.}\)
3. If \(aH=bH\) then \(Ha\) must equal \(Hb\text{.}\)
4. If \(a\in H\) then \(aH\) must equal \(Ha\text{.}\)
5. \(H\) must be normal in \(G\) if there exists \(a\in G\) such that \(aH=Ha\text{.}\)
6. If \(aH=bH\) then \(ah=bh\) for every \(h\in H\text{.}\)
7. If \(G\) is finite, then \(|G/H|\) must be less than \(|G|\text{.}\)
8. If \(G\) is finite, then \((G:H)\) must be less than or equal to \(|G|\text{.}\)

8. Let \(G\) be a group of order \(pq\text{,}\) where \(p\) and \(q\) are prime, and let \(H\) be a proper subgroup of \(G\text{.}\) Prove that \(H\) is cyclic.

9. Prove Corollary \(7.3.2\): that is, let \(G\) be a group of prime order, and prove that \(G\) is cyclic.