Math Problems Vault

Problem 62.1

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

Find the commutator:

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(a)

of the nonsingular matrices $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$ and $\left(\begin{array}{ll}a & 0 \\ 0 & 1\end{array}\right)$ ;

(b)

$\left(\begin{array}{ll}a & b \\ 0 & c\end{array}\right)$ and $\left(\begin{array}{ll}x & y \\ 0 & z\end{array}\right)$ ;

(c)

of two transpositions in the symmetric group $\mathbf{S}_{n}$ ;

(d)

$\left(\begin{array}{cc}\beta & 0 \\ 0 & \beta^{-1}\end{array}\right)$ and $\left(\begin{array}{ll}1 & \lambda \\ 0 & 1\end{array}\right)$ .

Problem 62.2

(a)(b)(c)

Prove the following properties of the commutator subgroup $G^{\prime }$ of a group:

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(a)

$G^{\prime }$ is a normal subgroup of $G$ ;

(b)

the quotient group $G / G^{\prime }$ is commutative;

(c)

if $N$ is normal in $G$ and $G / N$ is commutative, then $G^{\prime } \subseteq N$ .

Problem 62.3

Prove that under a surjective homomorphism $\varphi : G^{\prime } \rightarrow H$ the equality $\varphi (G)^{\prime }=H^{\prime }$ holds.

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Problem 62.4

Establish a bijective correspondence between homomorphisms of a group into commutative groups and homomorphisms of its quotient group by the commutator subgroup.

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Problem 62.5

Prove that the commutator subgroup of $\mathbf{G} \mathbf{L}_{n}(K)$ is contained in $\mathbf{S L}_{n}(K)$ .

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Problem 62.6

Prove that the commutator subgroup of a direct product is the direct product of the commutator subgroups of the factors.

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Problem 62.7

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

Find the commutator subgroups and the orders of the quotient groups by the commutator subgroups for the groups:

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(a)

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

(b)

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

(c)

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

(d)

$\mathbf{Q}_{8}$ .

Problem 62.8

(a)(b)

Find the commutator subgroups of the groups:

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(a)

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

(b)

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

Problem 62.9

Prove that the commutator subgroup of a normal subgroup is normal in the whole group.

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Problem 62.10

(a)(b)

The derived series (or commutator series) of a group $G$ is the series of subgroups

G=G^{0} \supseteq G^{\prime } \supseteq G^{\prime \prime } \supseteq \ldots ,

where $G^{i+1}=\left(G^{i}\right)^{\prime }$ . Prove that:

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(a)

all terms of the derived series are normal in $G$ ;

(b)

for every homomorphism $\varphi$ of the group $G$ onto a group $H$ , $\varphi \left(G^{i}\right)=H^{i}$ .

Problem 62.11

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

Prove that:

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(a)

every subgroup of a solvable group is solvable;

(b)

every quotient group of a solvable group is solvable;

(c)

if $A$ and $B$ are solvable groups, then the group $A \times B$ is solvable;

(d)

if $G / A \simeq B$ and $A, B$ are solvable groups, then $G$ is solvable.

Problem 62.12

Prove the solvability of the groups:

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(a)

$S_{3}$ ;

(b)

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

(c)

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

(d)

$\mathbf{Q}_{8}$ ;

(e)

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

Problem 62.13

(a)(b)(c)

Let $\mathbf{U T}_{n}(K)$ be the group of upper unitriangular matrices. Prove that:

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(a)

$\mathbf{U T}_{n}^{m}(K)$ (the set of matrices from $\mathbf{U T}_{n}(K)$ with $m-1$ zero diagonals above the main diagonal) is a subgroup of $\mathbf{U T}_{n}(K)$ ;

(b)

if $A \in \mathbf{U T}_{n}^{i}(K), B \in \mathbf{U T}_{n}^{j}(K)$ , then $[A, B] \in \mathbf{U T}_{n}^{(i+j)}(K)$ ;

(c)

the group $\mathbf{U T}_{n}(K)$ is solvable.

Problem 62.14

Prove that the group of nonsingular upper triangular matrices is solvable.

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Problem 62.15

Prove that a finite group $G$ is solvable if and only if it has a series of subgroups

G=H_{0} \supseteq H_{1} \supseteq \ldots \supseteq H_{k}=\left\{ e\right\}

such that $H_{i+1}$ is normal in $H_{i}$ and $H_{i} / H_{i+1}$ is a (cyclic) group of prime order.

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Problem 62.16

Prove that a finite $p$ -group is solvable.

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Problem 62.17

Prove the solvability of a group of order $p q$ , where $p, q$ are distinct prime numbers.

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Problem 62.18

Prove the solvability of a group of order:

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(a)

$20 ; \quad$

(b)

12 ;

(c)

$p^{2} q$ , where $p, q$ are distinct prime numbers;

(d)

42 ;

(e)

100 ;

(f)

$n<60$ .

Problem 62.19

Prove, for the transvections $t_{i j}(\alpha )=E+\alpha E_{i j}$ , the formula

\left[t_{i k}(\alpha ), t_{k j}(\beta )\right]=t_{i j}(\alpha \beta )

for distinct $i, j, k$ .

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Problem 62.20

(a)(b)

Let $F$ be a field and $n \geqslant 3$ . Prove that:

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(a)

$\mathbf{S L}_{n}^{\prime }(F)=\mathbf{G L}_{n}^{\prime }(F)=\mathbf{S L}_{n}(F)$ ;

(b)

the groups $\mathbf{S L}_{n}(F)$ and $\mathbf{G} \mathbf{L}_{n}(F)$ are not solvable.

Problem 62.21

(a)(b)

Let $F$ be a field containing at least four elements. Prove that:

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(a)

$\mathbf{S L}_{2}^{\prime }(F)=\mathbf{G L}_{2}^{\prime }(F)=\mathbf{S L}_{2}(F)$ ;

(b)

the groups $\mathbf{S L}_{2}(F)$ and $\mathbf{G} \mathbf{L}_{2}(F)$ are not solvable.

Problem 62.22

Let $p, q, r$ be distinct prime numbers. Prove that any group of order $p q r$ is solvable.

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Problem 62.23

Let $p, q, r$ be distinct prime numbers. Prove that a non-solvable group of order $p^{2} q r$ is isomorphic to $\mathbf{A}_{5}$ .

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Problem 62.24

If the order of a finite group is a product of distinct primes, then $G$ is a solvable group possessing a cyclic normal subgroup $N$ such that $G / N$ is cyclic.

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Problem 62.25

Let $G$ be a finite group such that $G=G^{\prime }$ , the center of $G$ has order 2, and the quotient group by the center is isomorphic to $\mathbf{A}_{5}$ . Prove that $G \simeq \mathbf{S L}_{2}\left(\mathbf{Z}_{5}\right)$ .

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Problem 62.26

(a)(b)(c)

Let $F$ be a field, $V$ an $n$ -dimensional vector space over $F$ , and $G$ a group of nonsingular linear operators on $V$ such that, if $g \in G$ , then $g=1+h$ , where $h^{n}=0$ . Prove that:

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(a)

there exists a vector $x \neq 0$ in $V$ such that $g x=x$ for all $g \in G$ ;

(b)

there exists a basis $e_{1}, \ldots , e_{n}$ of $V$ such that the matrices of all operators $g, g \in G$ , in this basis are upper triangular;

(c)

the group $G$ is solvable.

Problem 62.27

(a)(b)

Let $p, q$ be prime numbers such that $p$ divides $q-1$ . Prove that:

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(a)

there exists an integer $r \not\equiv 1(\bmod q)$ such that $r^{p} \equiv 1(\bmod q)$ ;

(b)

there exists (up to isomorphism) exactly one noncommutative group of order $p q$ .

Problem 62.28

(a)(b)(c)

Prove that:

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(a)

if in a commutative group the elements $a, b$ are related by $a^{3}=b^{5}=(a b)^{7}=e$ , then $a=b=e$ ;

(b)

the subgroup generated in $\mathbf{S}_{7}$ by the permutations (123) and (14567) is not solvable;

(c)

the group with generators $x_{1}, x_{2}$ and defining relations $x_{1}^{3}=x_{2}^{5}=\left(x_{1} x_{2}\right)^{7}=e$ is not solvable.

Problem 62.29

Is the free group solvable?

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