##### 1

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

\(G = S_4\) 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 \(D_4\). 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, \(Q_8\). 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 \({\mathbb R}\); that is, matrices of the form
\begin{equation*}\begin{pmatrix}
a & b \\
0 & c
\end{pmatrix},\end{equation*}
where \(a\), \(b\), \(c \in {\mathbb R}\) and \(ac \neq 0\). Let \(U\) consist of matrices of the form
\begin{equation*}\begin{pmatrix}
1 & x \\
0 & 1
\end{pmatrix},\end{equation*}
where \(x \in {\mathbb R}\).

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

Prove that \(U\) is abelian.

Prove that \(U\) is normal in \(T\).

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

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

##### 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 \(G\). Prove that \(H\) must be a normal subgroup of \(G\). Conclude that \(S_n\) is not simple for \(n \geq 3\).

##### 11

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

##### 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 \(G\). If \(g\) generates a normal subgroup of \(G\), prove that \(C(g)\) is normal in \(G\).

##### 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 \(S_3\).

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

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

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

##### 14

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

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

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