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Exercises 10.4 Exercises

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 \(G/H\text{.}\)
  1. \(G = S_4\) and \(H = A_4\)
  2. \(G = A_5\) and \(H = \{ (1), (1 \, 2 \, 3), (1 \, 3 \, 2) \}\)
  3. \(G = S_4\) and \(H = D_4\)
  4. \(G = Q_8\) and \(H = \{ 1, -1, I, -I \}\)
  5. \(G = {\mathbb Z}\) and \(H = 5 {\mathbb Z}\)
Hint.
(a)
\begin{equation*} \begin{array}{c|cc} & A_4 & (1 \, 2)A_4 \\ \hline A_4 & A_4 & (1 \, 2) A_4 \\ (1 \, 2) A_4 & (1 \, 2) A_4 & A_4 \end{array} \end{equation*}
(c) \(D_4\) is not normal in \(S_4\text{.}\)

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}\text{,} \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}\text{,} \end{equation*}
where \(x \in {\mathbb R}\text{.}\)
  1. Show that \(U\) is a subgroup of \(T\text{.}\)
  2. Prove that \(U\) is abelian.
  3. Prove that \(U\) is normal in \(T\text{.}\)
  4. Show that \(T/U\) is abelian.
  5. 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.
Hint.
If \(a \in G\) is a generator for \(G\text{,}\) then \(aH\) is a generator for \(G/H\text{.}\)

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 \(G\text{.}\)
Hint.
For any \(g \in G\text{,}\) show that the map \(i_g : G \to G\) defined by \(i_g : x \mapsto gxg^{-1}\) is an isomorphism of \(G\) with itself. Then consider \(i_g(H)\text{.}\)

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 \}\text{.} \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 \(G\text{.}\)
Hint.
Suppose that \(\langle g \rangle\) is normal in \(G\) and let \(y\) be an arbitrary element of \(G\text{.}\) If \(x \in C(g)\text{,}\) we must show that \(y x y^{-1}\) is also in \(C(g)\text{.}\) Show that \((y x y^{-1}) g = g (y x y^{-1})\text{.}\)

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 \}\text{.} \end{equation*}
  1. Calculate the center of \(S_3\text{.}\)
  2. Calculate the center of \(GL_2 ( {\mathbb R} )\text{.}\)
  3. Show that the center of any group \(G\) is a normal subgroup of \(G\text{.}\)
  4. 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 \(G\text{.}\)
  1. Show that \(G'\) is a normal subgroup of \(G\text{.}\)
  2. 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 \(G\text{.}\)
Hint.
(a) Let \(g \in G\) and \(h \in G'\text{.}\) If \(h = aba^{-1}b^{-1}\text{,}\) then
\begin{align*} ghg^{-1} & = gaba^{-1}b^{-1}g^{-1}\\ & = (gag^{-1})(gbg^{-1})(ga^{-1}g^{-1})(gb^{-1}g^{-1})\\ & = (gag^{-1})(gbg^{-1})(gag^{-1})^{-1}(gbg^{-1})^{-1}\text{.} \end{align*}
We also need to show that if \(h = h_1 \cdots h_n\) with \(h_i = a_i b_i a_i^{-1} b_i^{-1}\text{,}\) then \(ghg^{-1}\) is a product of elements of the same type. However, \(ghg^{-1} = g h_1 \cdots h_n g^{-1} = (gh_1g^{-1})(gh_2g^{-1}) \cdots (gh_ng^{-1})\text{.}\)
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