AATA Exercises

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Exercises 9.4 Exercises

1.

Prove that

Z ≅ n Z

for

n \neq 0 .

2.

Prove that

C^{*}

is isomorphic to the subgroup of

G L_{2} (R)

consisting of matrices of the form

(\begin{matrix} a & b \\ - b & a \end{matrix}) .

3.

Prove or disprove:

U (8) ≅ Z_{4} .

4.

Prove that

U (8)

is isomorphic to the group of matrices

(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}), (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}), (\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}), (\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}) .

5.

Show that

U (5)

is isomorphic to

U (10),

but

U (12)

is not.

6.

Show that the

n

th roots of unity are isomorphic to

Z_{n} .

7.

Show that any cyclic group of order

n

is isomorphic to

Z_{n} .

8.

Prove that

Q

is not isomorphic to

Z .

9.

Let

G = R ∖ {- 1}

and define a binary operation on

G

by

a * b = a + b + a b .

Prove that

G

is a group under this operation. Show that

(G, *)

is isomorphic to the multiplicative group of nonzero real numbers.

10.

Show that the matrices

\begin{array}{r} (\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}) (\begin{array}{c} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}) (\begin{array}{c} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}) \\ (\begin{array}{c} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}) (\begin{array}{c} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}) (\begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}) \end{array}

form a group. Find an isomorphism of

G

with a more familiar group of order

6 .

11.

Find five non-isomorphic groups of order

8 .

12.

Prove

S_{4}

is not isomorphic to

D_{12} .

13.

Let

ω = cis (2 π / n)

be a primitive

n

th root of unity. Prove that the matrices

A = (\begin{matrix} ω & 0 \\ 0 & ω^{- 1} \end{matrix}) and B = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})

generate a multiplicative group isomorphic to

D_{n} .

14.

Show that the set of all matrices of the form

(\begin{matrix} \pm 1 & k \\ 0 & 1 \end{matrix}),

is a group isomorphic to

D_{n},

where all entries in the matrix are in

Z_{n} .

15.

List all of the elements of

Z_{4} \times Z_{2} .

16.

Find the order of each of the following elements.

$(3, 4)$ in $Z_{4} \times Z_{6}$
$(6, 15, 4)$ in $Z_{30} \times Z_{45} \times Z_{24}$
$(5, 10, 15)$ in $Z_{25} \times Z_{25} \times Z_{25}$
$(8, 8, 8)$ in $Z_{10} \times Z_{24} \times Z_{80}$

17.

Prove that

D_{4}

cannot be the internal direct product of two of its proper subgroups.

18.

Prove that the subgroup of

Q^{*}

consisting of elements of the form

2^{m} 3^{n}

for

m, n \in Z

is an internal direct product isomorphic to

Z \times Z .

19.

Prove that

S_{3} \times Z_{2}

is isomorphic to

D_{6} .

Can you make a conjecture about

D_{2 n} ?

Prove your conjecture.

20.

Prove or disprove: Every abelian group of order divisible by

3

contains a subgroup of order

3 .

21.

Prove or disprove: Every nonabelian group of order divisible by 6 contains a subgroup of order

6 .

22.

Let

G

be a group of order

20 .

If

G

has subgroups

H

and

K

of orders

4

and

5

respectively such that

h k = k h

for all

h \in H

and

k \in K,

prove that

G

is the internal direct product of

H

and

K .

23.

Prove or disprove the following assertion. Let

G,

H,

and

K

be groups. If

G \times K ≅ H \times K,

then

G ≅ H .

24.

Prove or disprove: There is a noncyclic abelian group of order

51 .

25.

Prove or disprove: There is a noncyclic abelian group of order

52 .

26.

Let

ϕ : G \to H

be a group isomorphism. Show that

ϕ (x) = e_{H}

if and only if

x = e_{G},

where

e_{G}

and

e_{H}

are the identities of

G

and

H,

respectively.

27.

Let

G ≅ H .

Show that if

G

is cyclic, then so is

H .

28.

Prove that any group

G

of order

p,

p

prime, must be isomorphic to

Z_{p} .

29.

Show that

S_{n}

is isomorphic to a subgroup of

A_{n + 2} .

30.

Prove that

D_{n}

is isomorphic to a subgroup of

S_{n} .

31.

Let

ϕ : G_{1} \to G_{2}

and

ψ : G_{2} \to G_{3}

be isomorphisms. Show that

ϕ^{- 1}

and

ψ \circ ϕ

are both isomorphisms. Using these results, show that the isomorphism of groups determines an equivalence relation on the class of all groups.

32.

Prove

U (5) ≅ Z_{4} .

Can you generalize this result for

U (p),

where

p

is prime?

33.

Write out the permutations associated with each element of

S_{3}

in the proof of Cayley’s Theorem.

34.

An automorphism of a group

G

is an isomorphism with itself. Prove that complex conjugation is an automorphism of the additive group of complex numbers; that is, show that the map

ϕ (a + b i) = a - b i

is an isomorphism from

C

to

C .

35.

Prove that

a + i b \mapsto a - i b

is an automorphism of

C^{*} .

36.

Prove that

A \mapsto B^{- 1} A B

is an automorphism of

S L_{2} (R)

for all

B

in

G L_{2} (R) .

37.

We will denote the set of all automorphisms of

G

by

Aut (G) .

Prove that

Aut (G)

is a subgroup of

S_{G},

the group of permutations of

G .

38.

Find

Aut (Z_{6}) .

39.

Find

Aut (Z) .

40.

Find two nonisomorphic groups

G

and

H

such that

Aut (G) ≅ Aut (H) .

41.

Let

G

be a group and

g \in G .

Define a map

i_{g} : G \to G

by

i_{g} (x) = g x g^{- 1} .

Prove that

i_{g}

defines an automorphism of

G .

Such an automorphism is called an inner automorphism. The set of all inner automorphisms is denoted by

Inn (G) .

42.

Prove that

Inn (G)

is a subgroup of

Aut (G) .

43.

What are the inner automorphisms of the quaternion group

Q_{8} ?

Is

Inn (G) = Aut (G)

in this case?

44.

Let

G

be a group and

g \in G .

Define maps

λ_{g} : G \to G

and

ρ_{g} : G \to G

by

λ_{g} (x) = g x

and

ρ_{g} (x) = x g^{- 1} .

Show that

i_{g} = ρ_{g} \circ λ_{g}

is an automorphism of

G .

The isomorphism

g \mapsto ρ_{g}

is called the right regular representation of

G .

45.

Let

G

be the internal direct product of subgroups

H

and

K .

Show that the map

ϕ : G \to H \times K

defined by

ϕ (g) = (h, k)

for

g = h k,

where

h \in H

and

k \in K,

is one-to-one and onto.

46.

Let

G

and

H

be isomorphic groups. If

G

has a subgroup of order

n,

prove that

H

must also have a subgroup of order

n .

47.

If

G ≅ \overset{―}{G}

and

H ≅ \overset{―}{H},

show that

G \times H ≅ \overset{―}{G} \times \overset{―}{H} .

48.

Prove that

G \times H

is isomorphic to

H \times G .

49.

Let

n_{1}, \dots, n_{k}

be positive integers. Show that

\prod_{i = 1}^{k} Z_{n_{i}} ≅ Z_{n_{1} \dots n_{k}}

if and only if

gcd (n_{i}, n_{j}) = 1

for

i \neq j .

50.

Prove that

A \times B

is abelian if and only if

A

and

B

are abelian.

51.

If

G

is the internal direct product of

H_{1}, H_{2}, \dots, H_{n},

prove that

G

is isomorphic to

\prod_{i} H_{i} .

52.

Let

H_{1}

and

H_{2}

be subgroups of

G_{1}

and

G_{2},

respectively. Prove that

H_{1} \times H_{2}

is a subgroup of

G_{1} \times G_{2} .

53.

Let

m, n \in Z .

Prove that

⟨ m, n ⟩ = ⟨ d ⟩

if and only if

d = gcd (m, n) .

54.

Let

m, n \in Z .

Prove that

⟨ m ⟩ \cap ⟨ n ⟩ = ⟨ l ⟩

if and only if

l = lcm (m, n) .

55. Groups of order $2 p$ .

In this series of exercises we will classify all groups of order

2 p,

where

p

is an odd prime.

Assume $G$ is a group of order $2 p,$ where $p$ is an odd prime. If $a \in G,$ show that $a$ must have order $1,$ $2,$ $p,$ or $2 p .$
Suppose that $G$ has an element of order $2 p .$ Prove that $G$ is isomorphic to $Z_{2 p} .$ Hence, $G$ is cyclic.
Suppose that $G$ does not contain an element of order $2 p .$ Show that $G$ must contain an element of order $p .$ Hint: Assume that $G$ does not contain an element of order $p .$
Suppose that $G$ does not contain an element of order $2 p .$ Show that $G$ must contain an element of order $2 .$
Let $P$ be a subgroup of $G$ with order $p$ and $y \in G$ have order $2 .$ Show that $y P = P y .$
Suppose that $G$ does not contain an element of order $2 p$ and $P = ⟨ z ⟩$ is a subgroup of order $p$ generated by $z .$ If $y$ is an element of order $2,$ then $y z = z^{k} y$ for some $2 \leq k < p .$
Suppose that $G$ does not contain an element of order $2 p .$ Prove that $G$ is not abelian.
Suppose that $G$ does not contain an element of order $2 p$ and $P = ⟨ z ⟩$ is a subgroup of order $p$ generated by $z$ and $y$ is an element of order $2 .$ Show that we can list the elements of $G$ as ${z^{i} y^{j} ∣ 0 \leq i < p, 0 \leq j < 2} .$
Suppose that $G$ does not contain an element of order $2 p$ and $P = ⟨ z ⟩$ is a subgroup of order $p$ generated by $z$ and $y$ is an element of order $2 .$ Prove that the product $(z^{i} y^{j}) (z^{r} y^{s})$ can be expressed as a uniquely as $z^{m} y^{n}$ for some non negative integers $m, n .$ Thus, conclude that there is only one possibility for a non-abelian group of order $2 p,$ it must therefore be the one we have seen already, the dihedral group.

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