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Section10.3Exercises

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 \(G/H\).

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

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

  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 \(D_4\). 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 \(x \in {\mathbb R}\).

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

  2. Prove that \(U\) is abelian.

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

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

  5. 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*}

  1. Calculate the center of \(S_3\).

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

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

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

14

Let \(G\) be a group and let \(G' = \langle aba^{- 1} b^{-1} \rangle\); 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 \(G\).

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

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