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

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

\[ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix}\text{,} \nonumber \]

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

\[ \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}\text{,} \nonumber \]

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.

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

\[ C(g) = \{ x \in G : xg = gx \}\text{.} \nonumber \]

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

\[ Z(G) = \{ x \in G : xg = gx \text{ for all } g \in G \}\text{.} \nonumber \]

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 \(G\text{.}\)