##### 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!)

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!)

Let \(n\in \Z^+\text{.}\) Prove that \(\equiv_n\) is an equivalence relation on \(\Z\text{.}\)

The cells of the induced partition of \(\Z\) are called the

*residue classes*(or*congruence classes*)*of \(\Z\) modulo \(n\)*. Using set notation of the form \(\{\ldots,\#, \#,\#,\ldots\}\) for each class, write down the residue classes of \(\Z\) modulo \(4\text{.}\)

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

Find the indices of:

\(H=\langle (15)(24)\rangle\) in \(S_5\)

\(K=\langle (2354)(34)\rangle\) in \(S_6\)

\(A_n\) in \(S_n\)

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*.

\(H=6\Z\) in \(G=2\Z\)

\(H=\langle 4\rangle\) in \(\Z_{20}\)

\(H=\langle (23)\rangle\) in \(G=S_3\)

\(H=\langle r\rangle\) in \(G=D_4\)

\(H=\langle f\rangle\) in \(G=D_4\)

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.

A group \(G\) with subgroup \(H\) such that \(|G/H|=1\text{.}\)

A finite group \(G\) with subgroup \(H\) such that \(|G/H|=|G|\text{.}\)

An abelian group \(G\) of order \(8\) containing a non-normal subgroup \(H\) of order 2.

A group \(G\) of order 8 containing a normal subgroup of order \(2\text{.}\)

A nonabelian group \(G\) of order 8 containing a normal subgroup of index \(2\text{.}\)

A group \(G\) of order 8 containing a subgroup of order \(3\text{.}\)

An infinite group \(G\) containing a subgroup \(H\) of finite index.

An infinite group \(G\) containing a finite nontrivial subgroup \(H\text{.}\)

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 \(a,b\in G\text{.}\)

If \(a\in bH\) then \(aH\) must equal \(bH\text{.}\)

\(aH\) must equal \(Ha\text{.}\)

If \(aH=bH\) then \(Ha\) must equal \(Hb\text{.}\)

If \(a\in H\) then \(aH\) must equal \(Ha\text{.}\)

\(H\) must be normal in \(G\) if there exists \(a\in G\) such that \(aH=Ha\text{.}\)

If \(aH=bH\) then \(ah=bh\) for every \(h\in H\text{.}\)

If \(G\) is finite, then \(|G/H|\) must be less than \(|G|\text{.}\)

If \(G\) is finite, then \((G:H)\) must be less than or equal to \(|G|\text{.}\)

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.

Prove Corollary 7.3.10: that is, let \(G\) be a group of prime order, and prove that \(G\) is cyclic.