###### 1

For each of the following groups \(G\text{,}\) determine whether \(H\) is a normal subgroup of \(G\text{.}\) If \(H\) is a normal subgroup, write out a Cayley table for the factor group \(G/H\text{.}\)

\(G = S_4\) and \(H = A_4\)

\(G = A_5\) and \(H = \{ (1), (123), (132) \}\)

\(G = S_4\) and \(H = D_4\)

\(G = Q_8\) and \(H = \{ 1, -1, I, -I \}\)

\(G = {\mathbb Z}\) and \(H = 5 {\mathbb Z}\)

###### 2

Find all the subgroups of \(D_4\text{.}\) Which subgroups are normal? What are all the factor groups of \(D_4\) up to isomorphism?

###### 3

Find all the subgroups of the quaternion group, \(Q_8\text{.}\) Which subgroups are normal? What are all the factor groups of \(Q_8\) up to isomorphism?

###### 4

Let \(T\) be the group of nonsingular upper triangular \(2 \times 2\) matrices with entries in \({\mathbb R}\text{;}\) that is, matrices of the form

\begin{equation*}
\begin{pmatrix}
a & b \\
0 & c
\end{pmatrix},
\end{equation*}
where \(a\text{,}\) \(b\text{,}\) \(c \in {\mathbb R}\) and \(ac \neq 0\text{.}\) Let \(U\) consist of matrices of the form

\begin{equation*}
\begin{pmatrix}
1 & x \\
0 & 1
\end{pmatrix},
\end{equation*}
where \(x \in {\mathbb R}\text{.}\)

Show that \(U\) is a subgroup of \(T\text{.}\)

Prove that \(U\) is abelian.

Prove that \(U\) is normal in \(T\text{.}\)

Show that \(T/U\) is abelian.

Is \(T\) normal in \(GL_2( {\mathbb R})\text{?}\)

###### 5

Show that the intersection of two normal subgroups is a normal subgroup.

###### 6

If \(G\) is abelian, prove that \(G/H\) must also be abelian.

###### 7

Prove or disprove: If \(H\) is a normal subgroup of \(G\) such that \(H\) and \(G/H\) are abelian, then \(G\) is abelian.

###### 8

If \(G\) is cyclic, prove that \(G/H\) must also be cyclic.

###### 9

Prove or disprove: If \(H\) and \(G/H\) are cyclic, then \(G\) is cyclic.

###### 10

Let \(H\) be a subgroup of index \(2\) of a group \(G\text{.}\) Prove that \(H\) must be a normal subgroup of \(G\text{.}\) Conclude that \(S_n\) is not simple for \(n \geq 3\text{.}\)

###### 11

If a group \(G\) has exactly one subgroup \(H\) of order \(k\text{,}\) prove that \(H\) is normal in \(G\text{.}\)

###### 12

Define the **centralizer** of an element \(g\) in a group \(G\) to be the set

\begin{equation*}
C(g) = \{ x \in G : xg = gx \}.
\end{equation*}
Show that \(C(g)\) is a subgroup of \(G\text{.}\) If \(g\) generates a normal subgroup of \(G\text{,}\) prove that \(C(g)\) is normal in \(G\text{.}\)

###### 13

Recall that the **center** of a group \(G\) is the set

\begin{equation*}
Z(G) = \{ x \in G : xg = gx \text{ for all } g \in G \}.
\end{equation*}
Calculate the center of \(S_3\text{.}\)

Calculate the center of \(GL_2 ( {\mathbb R} )\text{.}\)

Show that the center of any group \(G\) is a normal subgroup of \(G\text{.}\)

If \(G / Z(G)\) is cyclic, show that \(G\) is abelian.

###### 14

Let \(G\) be a group and let \(G' = \langle aba^{- 1} b^{-1} \rangle\text{;}\) that is, \(G'\) is the subgroup of all finite products of elements in \(G\) of the form \(aba^{-1}b^{-1}\text{.}\) The subgroup \(G'\) is called the **commutator subgroup** of \(G\text{.}\)

Show that \(G'\) is a normal subgroup of \(G\text{.}\)

Let \(N\) be a normal subgroup of \(G\text{.}\) Prove that \(G/N\) is abelian if and only if \(N\) contains the commutator subgroup of \(G\text{.}\)

######
15Sage Exercise 1

Build every subgroup of the alternating group on 5 symbols, \(A_5\text{,}\) and check that each is not a normal subgroup (except for the two trivial cases). This command might take a couple seconds to run. Compare this with the time needed to run the `.is_simple()` method and realize that there is a significant amount of theory and cleverness brought to bear in speeding up commands like this. (It is possible that your Sage installation lacks GAP's “Table of Marks” library and you will be unable to compute the list of subgroups.)

######
16Sage Exercise 2

Consider the quotient group of the group of symmetries of an \(8\)-gon, formed with the cyclic subgroup of order \(4\) generated by a quarter-turn. Use the `coset_product` function to determine the Cayley table for this quotient group. Use the number of each coset, as produced by the `.cosets()` method as names for the elements of the quotient group. You will need to build the table “by hand” as there is no easy way to have Sage's Cayley table command do this one for you. You can build a table in the Sage Notebook pop-up editor (shift-click on a blue line) or you might read the documentation of the `html.table()` method.

######
17Sage Exercise 3

Consider the cyclic subgroup of order \(4\) in the symmetries of an \(8\)-gon. Verify that the subgroup is normal by first building the raw left and right cosets (without using the `.cosets()` method) and then checking their equality in Sage, all with a single command that employs sorting with the `sorted()` command.

######
18Sage Exercise 4

Again, use the same cyclic subgroup of order \(4\) in the group of symmetries of an \(8\)-gon. Check that the subgroup is normal by using part (2) of Theorem 9.3. Construct a one-line command that does the complete check and returns `True`. Maybe sort the elements of the subgroup `S` first, then slowly build up the necessary lists, commands, and conditions in steps. Notice that this check does not require ever building the cosets.

######
19Sage Exercise 5

Repeat the demonstration from the previous subsection that for the symmetries of a tetrahedron, a cyclic subgroup of order \(3\) results in an undefined coset multiplication. Above, the default setting for the `.cosets()` method builds right cosets — but in this problem, work instead with left cosets. You need to choose two cosets to multiply, and then demonstrate two choices for representatives that lead to different results for the product of the cosets.

######
20Sage Exercise 6

Construct some dihedral groups of order \(2n\) (i.e. symmetries of an \(n\)-gon, \(D_{n}\) in the text, `DihedralGroup(n)` in Sage). Maybe all of them for \(3\leq n \leq 100\text{.}\) For each dihedral group, construct a list of the orders of each of the normal subgroups (so use `.normal_subgroups()`). You may need to wait ten or twenty seconds for this to finish - be patient. Observe enough examples to hypothesize a pattern to your observations, check your hypothesis against each of your examples and then state your hypothesis clearly.

Can you predict how many normal subgroups there are in the dihedral group \(D_{470448}\) without using Sage to build all the normal subgroups? Can you *describe* all of the normal subgroups of a dihedral group in a way that would let us predict all of the normal subgroups of \(D_{470448}\) without using Sage?