Math Problems Vault

Problem 58.1

Prove that the subgroup $H$ of the group $G$ is normal, if:

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(a)

$G$ is a commutative group, $H$ is any subgroup of it;

(b)

$G=\mathbf{G} \mathbf{L}_{n}(\mathbb {R}), H$ is the subgroup of matrices with determinant equal to 1;

(c)

$G=\mathbf{S}_{n}, \quad H=\mathbf{A}_{n}$ ;

(d)

$G=\mathbf{S}_{4}, \quad H=\mathbf{V}_{4}$ ;

(e)

$G$ is the group of nonsingular complex upper triangular matrices, $H$ is the group of matrices of the form

E+\sum _{\substack {1 \leqslant i<j \leqslant n \\ j-i \geqslant k}} \alpha _{i j} E_{i j}, \quad \alpha _{i j} \in \mathbb {C}

Problem 58.2

Is the set of all matrices of the form

\left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right]

where $a, d$ are odd and $b, c$ are even, a normal subgroup of the group $\mathbf{G L}_{n}(\mathbb {Z})$ ?

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Problem 58.3

Prove that any subgroup of index 2 is normal.

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Problem 58.4

(a)(b)(c)

Find all normal subgroups, other than the trivial subgroup and the whole group, in the groups:

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(a)

$\mathbf{S}_{3}$ ;

(b)

$\mathbf{A}_{4}$ ;

(c)

$S_{4}$ .

Problem 58.5

Using the group $\mathbf{A}_{4}$ as an example, show that a normal subgroup $K$ of a normal subgroup $H$ of a group $G$ need not be normal in $G$ .

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Problem 58.6

Let $A$ and $B$ be normal subgroups of a group $G$ with $A \cap B$ the trivial subgroup. Prove that $x y=y x$ for all $x \in A, y \in B$ .

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Problem 58.7

Let $H$ be a subgroup of $G$ of index 2, $C$ a conjugacy class of $G$ , and $C \subset H$ . Prove that $C$ is either a single conjugacy class of $H$ , or the union of two conjugacy classes of $H$ consisting of the same number of elements.

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Problem 58.8

Prove that the quotient group $\mathbb {R}^{*} / \mathbb {Q}^{*}$ is not cyclic.

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Problem 58.9

Find the number of conjugacy classes in the group $\mathbf{A}_{5}$ and the number of elements in each class.

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Problem 58.10

Prove that the group $\mathbf{A}_{5}$ is simple.

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Problem 58.11

(a)(b)(c)

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(a)

Prove that in the quaternion group $\mathbf{Q}_{8}$ every subgroup is normal.

(b)

Find the center and all conjugacy classes in the group $\mathbf{Q}_{8}$ .

(c)

Prove that the complex matrices

\begin{array}{ll} \pm E= \pm \left[\begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{array}

, I= i 0 0 -i J= 0 1 -1 0 , K= 0 i i 0

form, under matrix multiplication, a group isomorphic to $\mathbf{Q}_{8}$ .

Problem 58.12

Find all normal subgroups of the dihedral group $\mathbf{D}_{n}$ .

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Problem 58.13

Let $F$ be a field and $G$ a subgroup of $\mathbf{G} \mathbf{L}_{n}(F)$ containing $\mathbf{S L}_{n}(F)$ . Prove that $G$ is normal in $\mathbf{G} \mathbf{L}_{n}(F)$ .

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Problem 58.14

To each matrix $\left(\begin{array}{ll}a & c \\ b & d\end{array}\right) \in \mathbf{G L}_{2}(\mathbb {C})$ assign the fractional-linear transformation

f(z)=\frac{a z+b}{c z+d}

Find the kernel of this homomorphism.

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Problem 58.15

Prove that the kernel of any homomorphism of the group $\mathbb {C}^{*}$ into the additive group $\mathbb {R}$ is an infinite group.

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Problem 58.16

(a)(b)(c)(d)

Let $n, m \geqslant 2$ be natural numbers, and let $\mathbf{S L}_{n}(\mathbb {Z} ; m \mathbb {Z})$ be the subset of $\mathbf{S L}_{n}(\mathbb {Z})$ consisting of matrices of the form $E+X m$ , where $X$ is an integer square matrix of size $n$ . Prove that:

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(a)

$\mathbf{S L}_{n}(\mathbb {Z} ; m \mathbb {Z})$ is a normal subgroup of $\mathbf{S L}_{n}(\mathbb {Z})$ ;

(b)

if $m=p$ is a prime number, then

\mathbf{S L}_{n}(\mathbb {Z}) / \mathbf{S} \mathbf{L}_{n}(\mathbb {Z} ; p \mathbb {Z}) \simeq \mathbf{S L}_{n}\left(\mathbb {Z}_{p}\right) ;

(c)

the group $\mathbf{S L}_{n}(\mathbb {Z} ; m \mathbb {Z})$ contains no elements of finite order when $m \geqslant 3$ ;

(d)

if $G$ is a finite subgroup of $\mathbf{S L}_{n}(\mathbb {Z})$ , then the order of $G$ divides

\frac{1}{2}\left(3^{n}-1\right)\left(3^{n}-3\right) \ldots \left(3^{n}-3^{n-1}\right)

Problem 58.17

Prove that for any group $G$ the set of all inner automorphisms is a normal subgroup of the group Aut $G$ of all automorphisms of $G$ .

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Problem 58.18

Prove that any subgroup containing the commutator subgroup of a group is normal.

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Problem 58.19

(a)(b)(c)

Find the center of the group:

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(a)

$\mathbf{S}_{n}$ ;

(b)

$\mathbf{A}_{n}$ ;

(c)

$\mathbf{D}_{n}$ .

Problem 58.20

Let the group $G$ be generated by elements $a, b$ , with

a^{2}=b^{2}=(a b)^{4}=1

Prove that the element $(a b)^{2}$ lies in the center of the group $G$ .

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Problem 58.21

Prove that the center of a group of order $p^{n}$ , where $p$ is a prime number $(n \in \mathbb {N})$ , contains more than one element.

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Problem 58.22

(a)(b)(c)

Let $G$ be the set of upper unitriangular matrices of order 3 with entries in the field $\mathbf{Z}_{p}$ .

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(a)

Prove that $G$ is a noncommutative group of order $p^{3}$ under multiplication.

(b)

Find the center of the group $G$ .

(c)

Find all the conjugacy classes of the group $G$ .

Problem 58.23

Find the center of the group:

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(a)

$\mathbf{G} \mathbf{L}_{n}(\mathbb {R})$ ;

(b)

$\mathbf{O}_{2}(\mathbb {R})$ ;

(c)

$\mathbf{S O}_{2}(\mathbb {R})$ ;

(d)

$\mathbf{S O}_{3}(\mathbb {R})$ ;

(e)

$\mathrm{SU}_{2}(\mathbb {C})$ ;

(f)

$\mathbf{S U}_{n}(\mathbb {C})$ ;

(g)

of the group of upper triangular matrices.

Problem 58.24

(a)(b)

Find the center:

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(a)

of the group of all fractional-linear transformations of the complex plane;

(b)

of the group of all transformations of the unit disk from problem 24.25.

Problem 58.25

Prove that a group $H$ is a homomorphic image of a finite cyclic group $G$ if and only if $H$ is also cyclic and its order divides the order of the group $G$ .

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Problem 58.26

(a)(b)

Prove that if a group $G$ is mapped homomorphically onto a group $H$ , with $a \mapsto a^{\prime }$ , then:

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(a)

the order of $a$ is divisible by the order of $a^{\prime }$ ;

(b)

the order of $G$ is divisible by the order of $H$ .

Problem 58.27

Find all homomorphic maps:

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(a)

$\mathbf{Z}_{6} \rightarrow \mathbf{Z}_{6}$ ;

(b)

$\mathbf{Z}_{6} \rightarrow \mathbf{Z}_{18}$ ;

(c)

$\mathbf{Z}_{18} \rightarrow \mathbf{Z}_{6}$ ;

(d)

$\mathbf{Z}_{12} \rightarrow \mathbf{Z}_{15}$ ;

(e)

$\mathbf{Z}_{6} \rightarrow \mathbf{Z}_{25}$ .

Problem 58.28

Prove that the additive group of rational numbers cannot be mapped homomorphically onto the additive group of integers.

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Problem 58.29

(a)(b)(c)(d)

Find the quotient groups:

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(a)

$\mathbb {Z} / n \mathbb {Z}$ ;

(b)

$\mathbf{U}_{12} / \mathbf{U}_{3}$ ;

(c)

$4 \mathbb {Z} / 12 \mathbb {Z}$ ;

(d)

$\mathbb {R}^{*} / \mathbb {R}_{+}$ .

Problem 58.30

Let $F^{n}$ be the additive group of an $n$ -dimensional linear space over a field $F$ , and let $H$ be the subgroup of vectors of a $k$ -dimensional subspace. Prove that the quotient group $F^{n} / H$ is isomorphic to $F^{n-k}$ .

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Problem 58.31

Let $H_{n}$ be the set of numbers with arguments of the form $2 \pi k / n$ $(k \in \mathbb {Z})$ . Prove that:

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(a)

$\mathbb {R} / \mathbb {Z} \simeq \mathrm{U}$ ;

(b)

$\mathbb {C}^{*} / \mathbb {R}^{*} \simeq \mathrm{U}$ ;

(c)

$\mathbb {C}^{*} / \mathbf{U} \simeq \mathbb {R}_{+}$ ;

(d)

$\mathbf{U} / \mathbf{U}_{n} \simeq \mathbf{U}$ ;

(e)

$\mathbb {C}^{*} / \mathrm{U}_{n} \simeq \mathbb {C}^{*}$ ;

(f)

$\mathbb {C}^{*} / H_{n} \simeq \mathbf{U}$ ;

(g)

$H_{n} / \mathbb {R}_{+} \simeq \mathbf{U}_{n}$ ;

(h)

$H_{n} / \mathbf{U}_{n} \simeq \mathbb {R}_{+}$ .

Problem 58.32

Let

\begin{aligned} G=\mathbf{G L}_{n}(\mathbb {R}), \quad H=\mathbf{G L}_{n}(\mathbb {C}), & & P=\mathbf{S L}_{n}(\mathbb {R}), \quad Q=\mathbf{S L}_{n}(\mathbb {C}) \\ A=\left\{ X \in G| | \operatorname {det} X \mid =1\right\} , & & B=\left\{ X \in H| | \operatorname {det} X \mid =1\right\} \\ B=\left\{ X \in G \mid \operatorname {det} X>1\right\} , & & N=\left\{ X \in H \mid \operatorname {det} X>0\right\} \end{aligned}

Prove that:

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(a)

$G / P \simeq \mathbb {R}^{*}$ ;

(b)

$H / Q \simeq \mathbb {C}^{*}$ ;

(c)

$G /(N \cap G) \simeq \mathbf{Z}_{2}$ ;

(d)

$H / N \simeq \mathbf{U}$ ;

(e)

$G / A \simeq \mathbb {R}_{+}$ ;

(f)

$H / B \simeq \mathbb {R}_{+}$ .

Problem 58.33

(a)(b)

Let $G$ be the group of affine transformations of an $n$ -dimensional space, $H$ the subgroup of translations, and $K$ the subgroup of transformations fixing a given point $O$ . Prove that:

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(a)

$H$ is a normal subgroup of $G$ ;

(b)

$G / H \simeq K$ .

Problem 58.34

Prove that the quotient group of $\mathbf{S}_{4}$ by the normal subgroup $\left\{ e,(12)(34),(13)(24),(14)(23)\right\}$ is isomorphic to the group $\mathbf{S}_{3}$ .

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Problem 58.35

Prove that if $H$ is a subgroup of index $k$ in a group $G$ , then $H$ contains a subgroup normal in $G$ whose index in $G$ divides $k!$ .

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Problem 58.36

Prove that a subgroup whose index is the smallest prime divisor of the order of the group is normal.

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Problem 58.37

Prove that the quotient group of $\mathbf{G} \mathbf{L}_{2}\left(\mathbf{Z}_{3}\right)$ by its center is isomorphic to the group $\mathbf{S}_{4}$ .

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Problem 58.38

(a)(b)

Prove that in the group $\mathbb {Q} / \mathbb {Z}$ :

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(a)

every element has finite order;

(b)

for every natural number $n$ there is exactly one subgroup of order $n$ .

Problem 58.39

Prove that the group of inner automorphisms of a group $G$ is isomorphic to the quotient group of $G$ by its center.

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Problem 58.40

Prove that the quotient group of a noncommutative group by its center cannot be cyclic.

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Problem 58.41

Prove that a group of order $p^{2}$ , where $p$ is a prime number, is commutative.

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Problem 58.42

Prove that the group of all automorphisms of a noncommutative group cannot be cyclic.

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Problem 58.43

Find the number of conjugacy classes and the number of elements in each class for a noncommutative group of order $p^{3}$ , where $p$ is a prime number.

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Problem 58.44

(a)(b)(c)

A subgroup $H$ is called maximal in a group $G$ if $H \neq G$ and any subgroup containing $H$ coincides with $H$ or $G$ . Prove that:

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(a)

the intersection of any two distinct maximal commutative subgroups is contained in the center of the group;

(b)

in every finite simple noncommutative group there exist two distinct maximal subgroups whose intersection contains more than one element;

(c)

in every finite simple noncommutative group there exists a proper noncommutative subgroup.

Problem 58.45

Prove that the quotient group of $\mathbf{S L}_{2}\left(\mathbf{Z}_{5}\right)$ by its center is isomorphic to $\mathbf{A}_{5}$ .

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Problem 58.46

Prove that every finite subgroup of $\mathrm{SL}_{2}(\mathbb {Q})$ is a subgroup of one of the following groups: $\mathbf{D}_{3}, \mathbf{D}_{4}, \mathbf{D}_{6}$ .

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Problem 58.47

Let $F$ be a field, $n \geqslant 3$ , and let $G$ be a normal subgroup of $\mathbf{G L}_{n}(F)$ . Prove that either $G \supseteq \mathbf{S L}_{n}(F)$ , or $G$ consists of scalar matrices.

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Problem 58.48

Let $F$ be a field containing at least four elements, and let $G$ be a normal subgroup of $\mathbf{G} \mathbf{L}_{2}(F)$ . Prove that either $G \supseteq \mathbf{S L}_{2}(F)$ , or $G$ consists of scalar matrices.

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Problem 58.49

Prove that $\mathbf{S L}_{2}(2) \simeq \mathbf{S}_{3}$ .

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Problem 58.50

Find all normal subgroups of $\mathbf{S L}_{2}(3)$ .

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Problem 58.51

Let $G$ be a normal subgroup of finite index in $\mathbf{S L}_{n}(\mathbb {Z}), n \geqslant 3$ . Then there exists a natural number $m$ such that $G \subseteq \mathbf{S L}_{n}(\mathbb {Z}, m \mathbb {Z})$ .

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Problem 58.52

Let $F$ be a field, $n \geqslant 3$ , and let $\varphi$ be an automorphism of the group $\mathbf{G L}_{n}(F)$ . Prove that there exist a group homomorphism $\tau : \mathbf{G} \mathbf{L}_{n}(F) \rightarrow F^{*}$ and an automorphism $\tau$ of the field $F$ such that either

\varphi (x)=\tau (x) g \tau (x) g^{-1},

or

\varphi (x)=\tau (x) g^{t} \tau (x)^{-1} g^{-1}

where $g \in \mathbf{G L}_{n}(F)$ .

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