### Homework 10, due Monday, May 7th:

**Do, but do not hand in:**

1. Prove that if \([E:F]=p\) where \(p\) is prime, then the only subfields of \(E\)
containing \(F\) are \(F\) and \(E\).

2. Determine all the subfields of:

2. Determine all the subfields of:

(a) \(\mathbb{Q}(\sqrt{7})\)

(b) \(\mathbb{Q}(\sqrt[3]{5})\)

(c) \(\mathbb{Q}(\sqrt[5]{11})\)

(d) \(\mathbb{Q}(\sqrt{7}, \sqrt{13})\)

3. Prove that \(\mathbb{Q}(\sqrt{2} + \sqrt{3}) = \mathbb{Q}(\sqrt{2}, \sqrt{3})\).(b) \(\mathbb{Q}(\sqrt[3]{5})\)

(c) \(\mathbb{Q}(\sqrt[5]{11})\)

(d) \(\mathbb{Q}(\sqrt{7}, \sqrt{13})\)

**Hand in:**

1. Prove that \(\mathbb{Q}(\sqrt{-3})\) is the splitting field for
\(x^3 - 1\) over \(\mathbb{Q}\).

2. Prove that \(\mathbb{Q}(\sqrt{7}, i)\) is the splitting field for \(x^4 - 6x^2 - 7\) over \(\mathbb{Q}\).

3. Let \(\theta\) be an element of an extension of a field \(F\), and let \(a, b \in F\) with \(a \neq 0\). Prove that \(F(\theta) = F(a \theta + b)\).

4. Let \(\theta = \sqrt{3} + \sqrt{-5} \in \mathbb{C}\).

2. Prove that \(\mathbb{Q}(\sqrt{7}, i)\) is the splitting field for \(x^4 - 6x^2 - 7\) over \(\mathbb{Q}\).

3. Let \(\theta\) be an element of an extension of a field \(F\), and let \(a, b \in F\) with \(a \neq 0\). Prove that \(F(\theta) = F(a \theta + b)\).

4. Let \(\theta = \sqrt{3} + \sqrt{-5} \in \mathbb{C}\).

(a) Prove that \([\mathbb{Q}(\theta):\mathbb{Q}] = 4\).

(b) Find the minimal polynomial for \(\theta\) over \(\mathbb{Q}\).

5. Let \(\alpha \in \mathbb{C}\) be algebraic over \(\mathbb{Q}\).
(b) Find the minimal polynomial for \(\theta\) over \(\mathbb{Q}\).

(a) Prove that \(\alpha^n\) is algebraic over \(\mathbb{Q}\)
for any \(n \in \mathbb{Z}\).

[Hint: Find an algebraic extension of \(\mathbb{Q}\) containing \(\alpha^n\).]

(b) Prove that \(\alpha^{1/q}\) is algebraic over \(\mathbb{Q}\) for any positive integer \(q\).

[Hint: Use the minimal polynomial for \(\alpha\) over \(\mathbb{Q}\) to construct a polynomial having \(\alpha^{1/q}\) as a zero.]

(c) Prove that \(\alpha^r\) is algebraic over \(\mathbb{Q}\) for any \(r \in \mathbb{Q}\).

[Hint: Use parts (a) and (b).]

[Hint: Find an algebraic extension of \(\mathbb{Q}\) containing \(\alpha^n\).]

(b) Prove that \(\alpha^{1/q}\) is algebraic over \(\mathbb{Q}\) for any positive integer \(q\).

[Hint: Use the minimal polynomial for \(\alpha\) over \(\mathbb{Q}\) to construct a polynomial having \(\alpha^{1/q}\) as a zero.]

(c) Prove that \(\alpha^r\) is algebraic over \(\mathbb{Q}\) for any \(r \in \mathbb{Q}\).

[Hint: Use parts (a) and (b).]

*Please typeset in LaTeX your solution to Exercise 5.*Include a statement of the problem. This exercise will be graded both on correctness and presentation.

### Homework 9, due Friday, April 20th:

**Do, but do not hand in:**

1. Prove that if \(D\) is an integral domain, \(a \in D\) is irreducible,
and \(u \in D\) is a unit, then \(au\) is irreducible.

**Hand in:**

1. Prove that \(\mathbb{Z}[\sqrt{-3}]\) is not a unique factorization domain.

[Hint: Use the properties of the norm \(N(a+b\sqrt{-3}) = |(a+b\sqrt{-3})(a-b\sqrt{-3})| = a^2 + 3b^2\).]

2. Let \(D\) be an integral domain, and let \(a,b \in D\). Prove that if \(a \in (b)\) and \(b \in (a)\), then \((a) = (b)\).

3. Let \(D\) be a principal ideal domain. Prove that an ideal in \(D\) is prime if and only if it's maximal.

[Hints: One direction is easy. For the other, observe that if \((p) \subseteq (q)\), then \(p = dq\) for some \(d \in D\). If \((p)\) is a prime ideal, \(p=dq \in (p)\) implies \(d \in (p)\) or \(q \in (p)\). Consider both cases and show that in one, \((q) = (p)\), and in the other, \((q) = D\).]

4. Give a construction of a finite field of order 16. List the elements of the field. Choose a nonzero, nonidentity element of the field and find its inverse.

Note: Either list the elements carefully, as cosets, or carefully explain whatever notation you use to simplify your description.

[Hint: Use the properties of the norm \(N(a+b\sqrt{-3}) = |(a+b\sqrt{-3})(a-b\sqrt{-3})| = a^2 + 3b^2\).]

2. Let \(D\) be an integral domain, and let \(a,b \in D\). Prove that if \(a \in (b)\) and \(b \in (a)\), then \((a) = (b)\).

3. Let \(D\) be a principal ideal domain. Prove that an ideal in \(D\) is prime if and only if it's maximal.

[Hints: One direction is easy. For the other, observe that if \((p) \subseteq (q)\), then \(p = dq\) for some \(d \in D\). If \((p)\) is a prime ideal, \(p=dq \in (p)\) implies \(d \in (p)\) or \(q \in (p)\). Consider both cases and show that in one, \((q) = (p)\), and in the other, \((q) = D\).]

4. Give a construction of a finite field of order 16. List the elements of the field. Choose a nonzero, nonidentity element of the field and find its inverse.

Note: Either list the elements carefully, as cosets, or carefully explain whatever notation you use to simplify your description.

*Please typeset in LaTeX your solution to Exercise 3.*Include a statement of the problem. This exercise will be graded both on correctness and presentation.

### Homework 8, due Wednesday, April 11th:

**Hand in:**

1. Let \(p(x) = x^3 + x^2 + x + 1\).

3. Prove that for every positive integer \(n\) there are infinitely many polynomials of degree \(n\) in \(\mathbb{Z}[x]\) that are irreducible over \(\mathbb{Q}\).

4. Let \(F\) be a field and let \(f(x) \in F[x]\) be reducible over \(F\). Prove that the ideal \((f(x))\) is not prime.

5. Test each polynomial below for reducibility over \(\mathbb{Q}\). If the polynomial is reducible, factor it completely. Otherwise, give a brief justification of why it is irreducible.

(a) Write \(p(x)\) as a product of irreducible polynomials over \(\mathbb{Q}\).

(b) Write \(p(x)\) as a product of irreducible polynomials over \(\mathbb{Z}/2\mathbb{Z}\).

2. Prove that \(x^2 - 5\) is irreducible over \(\mathbb{Q}\) but reducible over
\(\mathbb{Q}[\sqrt{5}]\).(b) Write \(p(x)\) as a product of irreducible polynomials over \(\mathbb{Z}/2\mathbb{Z}\).

3. Prove that for every positive integer \(n\) there are infinitely many polynomials of degree \(n\) in \(\mathbb{Z}[x]\) that are irreducible over \(\mathbb{Q}\).

4. Let \(F\) be a field and let \(f(x) \in F[x]\) be reducible over \(F\). Prove that the ideal \((f(x))\) is not prime.

5. Test each polynomial below for reducibility over \(\mathbb{Q}\). If the polynomial is reducible, factor it completely. Otherwise, give a brief justification of why it is irreducible.

(a) \(x^5 + 9x^4 -6x^3 +3x - 15\)

(b) \(x^3 + 7x^2 + 1\)

(c) \(x^4 + 5x^3 + 5\)

(d) \(x^3 + 3x^2 + 2\)

(b) \(x^3 + 7x^2 + 1\)

(c) \(x^4 + 5x^3 + 5\)

(d) \(x^3 + 3x^2 + 2\)

*Please typeset in LaTeX your solution to Exercise 4.*Include a statement of the problem. This exercise will be graded both on correctness and presentation.

### Homework 7, due Wednesday, April 4th:

**Notation:**

Let \(\mathbb{Z}/n\mathbb{Z}[x]\) denote the ring of all polynomials with
coefficients in \(\mathbb{Z}/n\mathbb{Z}\).

If \(a\in R\), the ideal generated by \(a\) is denoted \((a)\) (in contrast to the notation in your textbook: \(\langle a \rangle\)).

If \(a\in R\), the ideal generated by \(a\) is denoted \((a)\) (in contrast to the notation in your textbook: \(\langle a \rangle\)).

**Do, but do not hand in:**

1. List all polynomials of degree 2 in the ring \(\mathbb{Z}/2\mathbb{Z}[x]\).

2. Show that the polynomial \(2x+1\) has a multiplicative inverse in the ring \(\mathbb{Z}/4\mathbb{Z}[x]\).

3. Show that the polynomial \(x^2 + 3x + 2\) has four zeros in the ring \(\mathbb{Z}/6\mathbb{Z}[x]\).

2. Show that the polynomial \(2x+1\) has a multiplicative inverse in the ring \(\mathbb{Z}/4\mathbb{Z}[x]\).

3. Show that the polynomial \(x^2 + 3x + 2\) has four zeros in the ring \(\mathbb{Z}/6\mathbb{Z}[x]\).

**Hand in:**

1. Let \(a\) be an element of a ring \(R\). Define the

(You may refer to the previous homework set to assert that it's a subring.)

2. Prove that if \(A\) and \(B\) are ideals of a ring \(R\), then the set \[A + B = \{a + b \ : \ a \in A, b \in B\}\] is an ideal of \(R\).

3. Show that the quotient ring \(\mathbb{Z}/2\mathbb{Z}[x]/(x^2 + x + 1)\) is a field.

[Hint: Observe that \(x^2 = x+1\) in the quotient ring.]

4. Let \(R = \mathbb{Z}[x]\) and let \(I = (x^3 - 8)\).

*right annihilator*of \(a\) as the set \[\textrm{Ann}(a) = \{x \in R \ : \ ax = 0\}.\] Prove that \(\textrm{Ann}(a)\) is a right ideal of R.(You may refer to the previous homework set to assert that it's a subring.)

2. Prove that if \(A\) and \(B\) are ideals of a ring \(R\), then the set \[A + B = \{a + b \ : \ a \in A, b \in B\}\] is an ideal of \(R\).

3. Show that the quotient ring \(\mathbb{Z}/2\mathbb{Z}[x]/(x^2 + x + 1)\) is a field.

[Hint: Observe that \(x^2 = x+1\) in the quotient ring.]

4. Let \(R = \mathbb{Z}[x]\) and let \(I = (x^3 - 8)\).

(a) Find a polynomial of degree at most 2 that is congruent to
\[x^7 - x^6 + 5x^5 - 3x^4 + 2x^3 + x^2 - 6x + 2\]
in the quotient ring \(R/I\).

(b) Prove that \((x-2) + I\) is a zero divisor in \(R/I\).

(b) Prove that \((x-2) + I\) is a zero divisor in \(R/I\).

*Please typeset in LaTeX your solution to exercise 2.*Include a statement of the problem. This exercise will be graded both on correctness and presentation.

### Homework 6, due Wednesday, March 28th:

**Do, but do not hand in:**

1. Find a counterexample in a ring \(\mathbb{Z}/n\mathbb{Z}\) to each of the
following common principles of the algebra of real numbers:

(Hint: To prove that it's a ring, observe that \(\mathbb{Z}[\sqrt{5}] \subseteq \mathbb{R}\) and use the subring test.)

a) If \(ab = 0\), then \(a = 0\) or \(b = 0\).

b) If \(a^2 = a\), then \(a = 0\) or \(a = 1\).

c) If \(ab = ac\), then \(b = c\).

2. Prove that \(\mathbb{Z}[\sqrt{5}] = \{a + b\sqrt{5} \ : \ a,b \in \mathbb{Z}\}\)
is an integral domain.b) If \(a^2 = a\), then \(a = 0\) or \(a = 1\).

c) If \(ab = ac\), then \(b = c\).

(Hint: To prove that it's a ring, observe that \(\mathbb{Z}[\sqrt{5}] \subseteq \mathbb{R}\) and use the subring test.)

**Hand in:**

1. Let \(a\) be an element of a ring \(R\). Define the

2. Let \(R\) be a ring and let \(a \in R\). Prove that if \(a\) is a unit, then \(a\) is not a zero divisor.

3. Recall that \[\mathbb{Z} \oplus \mathbb{Q} = \{(z,q) \ : \ z \in \mathbb{Z} \textrm{ and } q \in \mathbb{Q}\}.\] Describe the set \(Z\) of all zero divisors and the set \(U\) of all units in the ring \(\mathbb{Z} \oplus \mathbb{Q}\).

4. Find a zero divisor in the ring \((\mathbb{Z}/5\mathbb{Z})[i] = \{a + bi \ : \ a,b \in \mathbb{Z}/5\mathbb{Z}\}\)

(where \(i\) is the usual imaginary unit, \(i = \sqrt{-1}\)).

5. Prove that \(\mathbb{Q}[\sqrt{5}] = \{a + b\sqrt{5} \ : \ a,b \in \mathbb{Q}\}\) is a field.

*right annihilator*of \(a\) as the set \[\textrm{Ann}(a) = \{x \in R \ : \ ax = 0\}.\] Prove that \(\textrm{Ann}(a)\) is a subring of R.2. Let \(R\) be a ring and let \(a \in R\). Prove that if \(a\) is a unit, then \(a\) is not a zero divisor.

3. Recall that \[\mathbb{Z} \oplus \mathbb{Q} = \{(z,q) \ : \ z \in \mathbb{Z} \textrm{ and } q \in \mathbb{Q}\}.\] Describe the set \(Z\) of all zero divisors and the set \(U\) of all units in the ring \(\mathbb{Z} \oplus \mathbb{Q}\).

4. Find a zero divisor in the ring \((\mathbb{Z}/5\mathbb{Z})[i] = \{a + bi \ : \ a,b \in \mathbb{Z}/5\mathbb{Z}\}\)

(where \(i\) is the usual imaginary unit, \(i = \sqrt{-1}\)).

5. Prove that \(\mathbb{Q}[\sqrt{5}] = \{a + b\sqrt{5} \ : \ a,b \in \mathbb{Q}\}\) is a field.

*Please typeset in LaTeX your solution to exercise 1.*Include a statement of the problem. This exercise will be graded both on correctness and presentation.

### Homework 5, due Wednesday, March 7th:

**Do, but do not hand in:**

1. Prove that if \(H\) and \(K\) are both normal subgroups of \(G\), then
\(H \cap K\) is a normal subgroup of \(G\).

**Hand in:**

1. A group of order \(p^n\) for some prime \(p\) and positive integer \(n\)
is called a

Let \(G\) be a non-abelian group of order \(p^3\) for some prime \(p\). Prove that \(|Z(G)| = p\).

(You will need the result stated above.)

2. Recall that if \(G_1\) and \(G_2\) are groups, the

Suppose \(\phi : Z_{30} \to Z_5\) is a nontrivial homomorphim. Find \(\ker \phi\).

3. Let \(H\) and \(K\) be subgroups of \(G\), and define \[HK = \{hk \ | \ h \in H \text{ and } k \in K\}.\]

*p-group*. We will not get to this result, but it can be shown that every \(p\)-group has a nontrivial center -- i.e. if \(G\) is a \(p\)-group, then \(Z(G) \neq 1\) (where "\(1\)" is the commonly used abbreviation for the identity subgroup, \(\{1\}\)).Let \(G\) be a non-abelian group of order \(p^3\) for some prime \(p\). Prove that \(|Z(G)| = p\).

(You will need the result stated above.)

2. Recall that if \(G_1\) and \(G_2\) are groups, the

*trivial homomorphism*from \(G_1\) to \(G_2\) is the mapping that sends every element of \(G_1\) to the identity element of \(G_2\). Any other homomorphism is called*nontrivial*.Suppose \(\phi : Z_{30} \to Z_5\) is a nontrivial homomorphim. Find \(\ker \phi\).

3. Let \(H\) and \(K\) be subgroups of \(G\), and define \[HK = \{hk \ | \ h \in H \text{ and } k \in K\}.\]

a) Let \(G \cong S_3\), \(H = \langle(1 \ 2)\rangle\), and
\(K = \langle(2 \ 3)\rangle\). Find \(HK\) and explain why \(HK\) is

b) Prove that if \(H \trianglelefteq G\), then \(HK\) is a subgroup of \(G\).

c) If we replace the condition "\(H \trianglelefteq G\)" with "\(K \leq N_G(H)\)," how must your argument in (b) change? State the theorem that results.

4. Let \(H\) and \(K\) be normal subgroups of \(G\), and suppose \(H \cap K = 1\)
(the identity subgroup) and \(G = HK\) (where \(HK\) is as defined in the previous
exercise).*not*a subgroup of \(G\).b) Prove that if \(H \trianglelefteq G\), then \(HK\) is a subgroup of \(G\).

c) If we replace the condition "\(H \trianglelefteq G\)" with "\(K \leq N_G(H)\)," how must your argument in (b) change? State the theorem that results.

a) Prove that the mapping \(\phi(hk) = k\) (where \(h \in H\) and \(k \in K\))
is a homomorphism from \(G\) to \(G\).

b) Use the First Isomorphism Theorem to prove that \(G/H \cong K\).

[Hint: Use part (a).]

b) Use the First Isomorphism Theorem to prove that \(G/H \cong K\).

[Hint: Use part (a).]

*Please typeset in LaTeX your solution to exercise 4.*Include a statement of the problem.

*This exercise will be graded both on correctness and on presentation.*

### Homework 4, due Friday, February 23rd:

**Do, but do not hand in:**

1. Find all of the left cosets of the subgroup \(H = \langle(1 \ 2 \ 3)\rangle\)
of \(A_4\).

2. How many left cosets of the subgroup \(H = \langle(1 \ 2 \ 3 \ 4)\rangle\) are there in \(S_6\)?

3. Suppose a group \(G\) contains elements of every order from 1 through 10. What is the minimum possible order of \(G\)?

2. How many left cosets of the subgroup \(H = \langle(1 \ 2 \ 3 \ 4)\rangle\) are there in \(S_6\)?

3. Suppose a group \(G\) contains elements of every order from 1 through 10. What is the minimum possible order of \(G\)?

**Hand in:**

**Notation:**For any subset \(S = \{x_1, x_2, \ldots, x_n\}\) of any group \(G\), we call the smallest subgroup containing \(S\) the

*subgroup generated by \(S\)*, denoted \(\langle x_1, x_2, \ldots, x_n\rangle\) or \(\langle S\rangle\). (This is an extension of the familiar notation for cyclic groups and subgroups, which are generated by a single element.)

1. Let \(p\) and \(q\) be distinct primes, and let \(G\) be a group of order \(pq\). Prove that if \(x\) and \(y\) are nonidentity elements of \(G\) and \(|x| \neq |y|\), then \(\langle x,y \rangle = G\).

2. Let \(p\) be a prime and let \(G\) be a group of order \(p^2\). Prove that either \(G\) is cyclic or \(g^p = 1\) for every \(g \in G\).

3. Consider the permutations \(\sigma = (1 \ 2)\) and \(\tau = (2 \ 3 \ 4)\) as elements of \(S_4\). Prove that \(\langle \sigma, \tau \rangle = S_4\).

4. Let \(H \leq G\). Prove that \(C_G(H) \trianglelefteq N_G(H)\).

(We proved in class that \(C_G(H) \leq N_G(H)\), so you need only prove that \(C_G(H)\) is

*normal*in \(N_G(H)\)).

*Please typeset in LaTeX your solution to exercise 4.*Include a statement of the problem.

*(This exercise will be graded both on correctness and on presentation.)*

### Homework 3, due Wednesday, February 14th:

**Do, but do not hand in:**

1. Prove that \(\left|x^{-1}\right| = \left|x\right|\) for any element \(x\)
of any group \(G\).

2. Let \(A = \begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}\) and \(B = \begin{bmatrix}0 & 1\\-1 & -1\end{bmatrix}\) in \(\textrm{SL}_2(\mathbb{R})\). Find \(|A|\), \(|B|\), and \(|AB|\).

3. Let \(G = \langle a \rangle\) and let \(|a| = 24\).

6. Determine whether each of the permutations in exercise 4 is even or odd.

7. Prove that \(A_8\) contains an element of order 15.

2. Let \(A = \begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}\) and \(B = \begin{bmatrix}0 & 1\\-1 & -1\end{bmatrix}\) in \(\textrm{SL}_2(\mathbb{R})\). Find \(|A|\), \(|B|\), and \(|AB|\).

3. Let \(G = \langle a \rangle\) and let \(|a| = 24\).

a) List all generators for the subgroup of order 8.

b) List all generators for \(G\).

4. Write each of the following as a product of disjoint cycles:b) List all generators for \(G\).

a) (1 2 3 4)(1 3 5)

b) (1 2 3 4)(1 2 3)

c) (1 5)(1 4)(1 3)(1 2)

d) (1 2 3 4)(1 3)

e) (1 2 3 4)(1 4 3)

f) (1 2 3 4)(3 4 5 6)

5. Find the order of each of the permutations in exercise 4.b) (1 2 3 4)(1 2 3)

c) (1 5)(1 4)(1 3)(1 2)

d) (1 2 3 4)(1 3)

e) (1 2 3 4)(1 4 3)

f) (1 2 3 4)(3 4 5 6)

6. Determine whether each of the permutations in exercise 4 is even or odd.

7. Prove that \(A_8\) contains an element of order 15.

**Hand in:**

1. Give the subgroup lattice for \(Z_{12} = \langle a \rangle\).

2. Let \(G = \langle a \rangle\) with \(|G| = n\). Prove that \(g^n = 1\) for every \(g \in G\).

3. Prove that \(S_n\) is non-abelian for all \(n \geq 3\), and that \(A_n\) is non-abelian for all \(n \geq 4\).

4. Let \(G\) be a group of permutations on a set \(\Omega\). Let \(a \in \Omega\), and define

2. Let \(G = \langle a \rangle\) with \(|G| = n\). Prove that \(g^n = 1\) for every \(g \in G\).

3. Prove that \(S_n\) is non-abelian for all \(n \geq 3\), and that \(A_n\) is non-abelian for all \(n \geq 4\).

4. Let \(G\) be a group of permutations on a set \(\Omega\). Let \(a \in \Omega\), and define

*the stabilizer in \(G\) of \(a\)*as: \[G_a = \{\sigma \in G \ : \ \sigma(a) = a\}.\] Prove that \(G_a\) is a subgroup of \(G\).*Note*: Gallian calls this subgroup "stab(\(a\))." His solution, which is probably in your textbook (and which you are welcome to look at), is both terse and incomplete. Make sure to give a complete solution in your own words.*Please typeset in LaTeX your solution to exercise 4. (This exercise will be graded for both correctness and presentation.)*You may use this LaTeX file as a template, if you like.

### Homework 2, due Monday, February 5th:

1. Let \(H\) and \(K\) be subgroups of \(G\).
Prove that \(H \cap K\) is a subgroup of \(G\).

2. Let \(H\) and \(K\) be subgroups of \(G\). Prove that \(H \cup K\) is a subgroup of \(G\) if and only if either \(H \subseteq K\) or \(K \subseteq H\).

3. Let \(a\) be an element of a group \(G\). Prove that \(C_G(a) = C_G(a^{-1})\).

4. Prove that if \(H \leq G\), \(K \leq G\), and \(H \subseteq K\), then \(H \leq K\).

5. Let \(H\) be a subgroup of \(G\). Prove that \(H \leq C_G(H)\) if and only if \(H\) is abelian.

6. Let \(G\) be a group.

2. Let \(H\) and \(K\) be subgroups of \(G\). Prove that \(H \cup K\) is a subgroup of \(G\) if and only if either \(H \subseteq K\) or \(K \subseteq H\).

3. Let \(a\) be an element of a group \(G\). Prove that \(C_G(a) = C_G(a^{-1})\).

4. Prove that if \(H \leq G\), \(K \leq G\), and \(H \subseteq K\), then \(H \leq K\).

5. Let \(H\) be a subgroup of \(G\). Prove that \(H \leq C_G(H)\) if and only if \(H\) is abelian.

6. Let \(G\) be a group.

(a) Let \(H \leq G\). Show that
\(C_G(H) = \{g \in g \ \mid \ g^{-1}hg = h \textrm{ for all } h \in H\}\).

(b) Let \(g, h \in G\). Prove that \((g^{-1}hg)^n = g^{-1}h^n g\) for any \(n \in \mathbb{N}\).

(c) Conclude that \(C_G(h) \leq C_G(h^n)\) for any \(n \in \mathbb{N}\).

(In other words,

(b) Let \(g, h \in G\). Prove that \((g^{-1}hg)^n = g^{-1}h^n g\) for any \(n \in \mathbb{N}\).

(c) Conclude that \(C_G(h) \leq C_G(h^n)\) for any \(n \in \mathbb{N}\).

(In other words,

*if \(g\) commutes with \(h\), then \(g\) commutes with all powers of \(h\)*.)### Homework 1, due Friday, 1/26:

1. Prove that the operation \(*\) defined on \(\mathbb{Q} - \{0\}\) by
\(a * b = \displaystyle \frac{a}{b}\) is not associative.

2. Prove that the operation \(*\) defined on \(\mathbb{Q}\) by \(a * b = a + b + ab\) is associative.

3. Suppose \(x,y,z \in G\) where \(G\) is a group (under the operation of multiplication). What is the inverse of the element \(xyz\)?

4. Suppose \(G\) is a group (under the operation of multiplication) with identity \(1\) such that \(x^2 = 1\) for every \(x \in G\). Prove that \(G\) is abelian.

5. Let \(G = \{a + b \sqrt{2} \mid a,b \in \mathbb{Q}\}\).

2. Prove that the operation \(*\) defined on \(\mathbb{Q}\) by \(a * b = a + b + ab\) is associative.

3. Suppose \(x,y,z \in G\) where \(G\) is a group (under the operation of multiplication). What is the inverse of the element \(xyz\)?

4. Suppose \(G\) is a group (under the operation of multiplication) with identity \(1\) such that \(x^2 = 1\) for every \(x \in G\). Prove that \(G\) is abelian.

5. Let \(G = \{a + b \sqrt{2} \mid a,b \in \mathbb{Q}\}\).

(a) Prove that \(G\) is a group under addition.

(b) Prove that \(G - \{0\}\) is a group under multiplication.

[For both parts, show that the operation is in fact a binary operation on \(G\) -- i.e. \(x, y \in G\) implies \(x*y \in G\). Also (for both parts), you can argue that associativity is inherited from \(\mathbb{R}\) (since \(G \subset \mathbb{R}\), \(x, y, z \in G\) implies \(x, y, z \in \mathbb{R}\)).]

(b) Prove that \(G - \{0\}\) is a group under multiplication.

[For both parts, show that the operation is in fact a binary operation on \(G\) -- i.e. \(x, y \in G\) implies \(x*y \in G\). Also (for both parts), you can argue that associativity is inherited from \(\mathbb{R}\) (since \(G \subset \mathbb{R}\), \(x, y, z \in G\) implies \(x, y, z \in \mathbb{R}\)).]