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Section 10.3 Exercises

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

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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?

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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?

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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 \(GL_2( {\mathbb R})\text{?}\)

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5

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

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6

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

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7

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

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8

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

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9

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

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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 \(n \geq 3\text{.}\)

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11

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

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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 \(G\text{.}\)

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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.

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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 \(G\text{.}\)