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 $H = 5 {\\mathbb Z}$
Hint(a)
\n\\begin{equation*}\n\\begin{array}{c|cc}\n& A_4 & (12)A_4 \\\\ \\hline\nA_4 & A_4 & (12) A_4 \\\\\n(12) A_4 & (12) A_4 & A_4\n\\end{array}\n\\end{equation*}\n
(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
\n\\begin{equation*}\n\\begin{pmatrix}\na & b \\\\\n0 & c\n\\end{pmatrix},\n\\end{equation*}\n
where $a\\text{,}$ $b\\text{,}$ $c \\in {\\mathbb R}$ and $ac \\neq 0\\text{.}$ Let $U$ consist of matrices of the form
\n\\begin{equation*}\n\\begin{pmatrix}\n1 & x \\\\\n0 & 1\n\\end{pmatrix},\n\\end{equation*}\n
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 $GL_2( {\\mathbb R})\\text{?}$
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.
HintIf $a \\in G$ 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 $G\\text{.}$
HintFor 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
\n\\begin{equation*}\nC(g) = \\{ x \\in G : xg = gx \\}.\n\\end{equation*}\n
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 $G\\text{.}$
HintSuppose 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
\n\\begin{equation*}\nZ(G) = \\{ x \\in G : xg = gx \\text{ for all } g \\in G \\}.\n\\end{equation*}\n
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.
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 $G\\text{.}$
Hint(a) Let $g \\in G$ and $h \\in G'\\text{.}$ If $h = aba^{-1}b^{-1}\\text{,}$ then
\n\\begin{align*}\nghg^{-1} & = gaba^{-1}b^{-1}g^{-1}\\\\\n& = (gag^{-1})(gbg^{-1})(ga^{-1}g^{-1})(gb^{-1}g^{-1})\\\\\n& = (gag^{-1})(gbg^{-1})(gag^{-1})^{-1}(gbg^{-1})^{-1}.\n\\end{align*}\n
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, $ghg^{-1} = g h_1 \\cdots h_n g^{-1} = (gh_1g^{-1})(gh_2g^{-1}) \\cdots (gh_ng^{-1})\\text{.}$