Advanced Algebra 1, 2022 Semester 1

The mid-semester and the final exam are hurdles. A student who fails any of these will receive an NCN.

If you are taking the ASE, it will count towards 25% of the marks. If you are registered under the code MATH6118, the ASE is mandatory.

Verify that \(G_p \subset G\) is a subgroup.

Let \(G = D_n\), the group of symmetries of a regular \(n\)-gon. Find the orbits and stabilisers of various points. How many different kinds of stabilisers do you see?

Let \(G\) be the group of isometries of one of the wallpaper patterns on the next page. Find the orbits and stabilisers of various points \(p\) in the plane. How many different kinds of stabilisers do you see?

Let \(G\) be as before and let \(H\) be the associated point group. Prove that the map \(\phi \colon G_p \to H\) is injective. Can you always find a point \(p\) for which the map is also surjective?

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Find the conjugacy classes and hence the class equation for \(D_{5}\) and \(D_{6}\). Generalise your results to \(D_{n}\).

There are 3 ways to partition \(\{1,2,3,4\}\) into 2 subsets of size 2. Use this to construct a non-trivial homomorphism \(\phi \colon S_4 \to S_3\). What is the kernel of \(\phi\)?

Fix positive integers \(n\) and \(k\) with \(k \leq n\) (for example, \(n = 4\) and \(k = 2\)). Let \(\operatorname{Gr}(k,n)\) be the set of \(k\)-dimensional sub-spaces of \(\mathbf{R}^n\). The natural action of \(\operatorname{GL}_n(\mathbf{R})\) on \(\mathbf{R}^{n}\) induces an action on \(\operatorname{Gr}(k,n)\).

1. Let \(V\) be the subspace of vectors where the last \(n-k\) co-ordinates are 0. Find the stabiliser of \(V\).
2. Show that the action is transitive.
3. Let \(S\) be the set of all two element subsets of \(\operatorname{Gr}(k,n)\). Is the action of \(\operatorname{GL}_n(\mathbf{R})\) on \(S\) transitive? What are the orbits of the action?

1. Let \(M = \mathbf{Z}/2 \mathbf{Z} \oplus \mathbf{Z}/4 \mathbf{Z}\) considered as a \(\mathbf{Z}\)-module. Let \(g_1 = (1,1), g_2 = (1,2), g_3=(0,1)\). Do \(g_{1}, g_2, g_3\) generate \(M\)?
2. Consider the map \(\mathbf{Z^3} \to M\). Find a set of generators for its kernel.

1. Let \(N\) be the module generatored by \(x, y\) modulo the relations \[ x + 2y = 0, \quad y + 2x = 0.\] Express \(N\) as the cokernel of a matrix.
2. Does \(z = x+y\) generate \(N\)?
3. What is the kernel of the map \(\mathbf{Z} \to N\) that sends \(1 \to z\)?
4. Describe \(N\) up to isomorphism in as simple terms as possible.

1. Let \(P = \mathbf{C}\) considered as a \(R = \mathbf{C}[x,y]\)-module where \(x\) and \(y\) act by \(0\). Find a set of generators for \(P\).
2. Let \(R^n \to P\) be the surjection given by your generators. Find the kernel \(P'\), and a set of generators for \(P'\).
3. Repeat, until you reach the 0 module.

The process of repeatedly finding generators and relations is called finding a resolution. For some rings, all finitely generated modules have finite resolutions, but not for all.

• For review: Artin, Chapter 1, Sections 1.4, 1.5
• Artin, Chapter 2, Sections 2.1, 2.2, 2.3, 2.4

• Artin, Chapter 2, Sections 2.5, 2.6, 2.7, 2.8

(Artin 11.4, modified) Given subgroups \(H, K \subset G\), we have a map \( \phi \colon H \times K \to G\) given by \(\phi(h,k) = hk\). Read Proposition 2.11.4 (Artin) for the basic properties of this map. In each of the following cases, determine whether or not \(\phi \colon H \times K \to G\) is an isomorphism

1. \(G = \mathbf{R}^{\times}\), \(H = \{\pm 1\}\), \(K = \mathbf{R}_{>0}^{\times}\).
2. \(G = \{\text{invertible upper triangular $2 \times 2$ matrices}\}\), \(H = \{\text{invertible diagonal matrices}\}\),\newline \(K = \{\text{upper triangular matrices with diagonal entries 1}\}\).

(Artin 12.4, modified) Let \(G = \mathbf{C}^{\times}\) and let \(H \subset G\) be the subgroup consisting of the \(n\)-th roots of unity: \[ H = \{\zeta \in G \mid \zeta^{n} = 1\}.\] For \(n = 4\), describe the cosets of \(H\) in \(G\) explicitly. For any \(n\), construct an isomorphism \(G/H \to G\).

Use the correspondance theorem for subgroups to describe all subgroups of \(\mathbf{Z}/d \mathbf{Z}\). Using your description, show that for every positive integer \(e\) dividing \(d\), there is a unique subgroup of \(\mathbf{Z}/d \mathbf{Z}\) of order \(e\).

Let \(N \subset G\) be a normal subgroup. Give an example where \(G\) is isomorphic to \(N \times G/N\) and an example where \(G\) is not isomorphic to \(N \times G/N\). Justify your examples.

(Artin, Chapter 6, 4.2a, modified). Find all subgroups of the dihedral group \(D_4\) and determine which ones are normal. For the normal ones, identify the quotient group up to isomorphism.

Hint: You may use that any subgroup of index 2 is normal; see (Chapter 2, Problem 8.10).

(Artin, Chapter 2, M.7, modified). Let \(G\) be a subgroup of \(\operatorname{GL}_n(\mathbf{R})\). On \(G\), define a relation \(x \sim y\) if there exists a continuous map \(\phi \colon [0,1] \to G\) with \(\phi(0) = x\) and \(\phi(1) = y\). Convince yourself (but do not write/submit) that \(\sim\) is an equivalence relation.

Let \(G_0 \subset G\) be the set of elements equivalent to the identity. Prove that \(G_0\) is a normal subgroup of \(G\). (It is called the connected component of the identity.)

Let \(G = O_2\). Find \(G_0\) with proof.

Find, with justification, the point group of the following pattern (assumed to extend infinitely). For each element of the point group, identify an isometry that maps to it.

(Artin, Chapter 7, Problem 2.1) Determine the centralizer and the order of the conjugacy class of

1. the matrix \(\begin{pmatrix}1 & 1 \\ & 1 \end{pmatrix}\) in \(\operatorname{GL}_2(\mathbf{F}_3),\)
2. the matrix \(\begin{pmatrix}1 & \\ & 2 \end{pmatrix}\) in \(\operatorname{GL}_2(\mathbf{F}_5).\)

Let \(G\) be a group of order \(2022\). Prove that \(G\) cannot be simple; that is, it must have a normal subgroup other than the trivial subgroup and the entire group.

(Artin, Chapter 7, Problem 7.8, modified) Let \(G = \operatorname{GL}_2(\mathbf{F}_p)\). Find the order of \(G\). Find a Sylow \(p\)-subgroup of \(G\) and determine the number of Sylow \(p\)-subgroups.

(Artin, Chapter 7, Problem M.1) Classify groups that are generated by two elements \(x\) and \(y\) of order 2.
Hint: It will be convenient to make use of the element \(z = xy\).

“Classify” means “find all up to isomorphism”.

(Artin, Chapter 11, Problem 1.6, modified) Decide (with justification) whether or not \(S\) is a subring of \(R\) where

1. \(R = \mathbf{Q}\) and \(S \subset R\) is the set of rational numbers \(a/b\) where \(b\) is not divisible by 3.
2. \(R = \text{the set of real valued functions on $\mathbf{R}$}\), and \(S \subset R\) is the set of bounded functions.

Construct a ring homomorphism or argue that no such homomorphism exists:

1. \(\mathbf{Z}/5 \mathbf{Z} \to \mathbf{Q}\)
2. \(\mathbf{Q} \to \mathbf{Z}/ 5 \mathbf{Z}\)
3. \(\mathbf{Z}[x,y]/(x^2+y^2-1) \to \mathbf{Q}\)
4. \(\mathbf{Z}[x,y]/(x^2+y^2-3) \to \mathbf{Z}\)

(Artin, Chapter 11, Problem 1.8, modified) Determine (with proof) the units in \(\mathbf{Z} / n \mathbf{Z}\).

Let \(R\) be a ring. An \(n \times n\) matrix \(A\) with entries in \(R\) is invertible over \(R\) if there is an \(n \times n\) matrix \(B\) with entries in \(R\) such that \(AB = BA = \operatorname{id}\). Prove for yourself (but do not turn in) the following theorem:

Theorem: A matrix \(A\) is invertible over \(R\) if and only if \(\det A\) is a unit in \(R\).

1. Prove that a ring homomorphism \(\phi \colon R \to S\) induces a group homomorphism \[ \operatorname{GL}_n(R) \to \operatorname{GL}_n(S),\] obtained by applying \(\phi\) to each entry of the matrix.
2. Let \(p\) be a prime and let \(\phi \colon \mathbf{Z}/ p^2 \mathbf{Z} \to \mathbf{Z} / p \mathbf{Z}\) be the unique ring homomorphism. Prove that the induced homomorphism \[ \operatorname{GL}_n(\mathbf{Z}/p^2 \mathbf{Z}) \to \operatorname{GL}_n(\mathbf{Z} / p \mathbf{Z})\] is surjective and its kernel is isomorphic to the additive group \((\mathbf{Z} / p \mathbf{Z})^{n^2}\).
3. (Not to be turned in) Generalise the statement above to \(\operatorname{GL}_n(\mathbf{Z}/p^m \mathbf{Z})\).

A principal ideal domain is an integral domain \(R\) in which every ideal is of the form \(fR\) for some \(f \in R\). For example, \(\mathbf{Z}\) is a principal ideal domain. Prove that \(\mathbf{Z}[x]\) is not a principal ideal domain.

Remark: In a principal ideal domain \(R\), we have a reasonable definition of gcd. Given \(f, g \in R\), we consider the ideal \((f,g)R\), which must be of the form \(hR\) for some \(h \in R\). This \(h\) is not necessarily unique, but it is unique up to multiplying by a unit, and we call it the gcd of \(f\) and \(g\). But if \(R\) is not a principal ideal domain, then there may not be any such \(h\).

(Artin, Chapter 11, 3.8) Let \(p\) be a prime number and let \(R\) be a ring in which \(p = 0\). Prove that the map \(R \to R\) that sends \(x\) to \(x^p\) is a ring homomorphism.

Remark: This homomorphism is called the Frobenius homomorphism. The Frobenius is the identity map for \(R = \mathbf{Z}/p \mathbf{Z}\), but in general, it is different from the identity. In fact, it may not even be an isomorphism (example: \(R = \mathbf{F}_p[x]\)).

(Artin, Chapter 11, 4.4, modified) Are the rings \(\mathbf{Z}[x]/(x^2+7)\) and \(\mathbf{Z}[x]/(2x^2+7)\) isomorphic? Does your answer change if you change \(\mathbf{Z}\) to \(\mathbf{C}\)?

Find generators for the kernel of the following maps:

1. \(\mathbf{Z}[x] \to \mathbf{F}_7[t]/(t^2+1)\) that sends \(x \mapsto 2t\).
2. \(\mathbf{C}[x,y] \to \mathbf{C}[t]\) which is the identity on \(\mathbf{C}\) and sends \(x \to t^2, y \mapsto t^3\).

(Artin Chapter 11, 3.9(a), modified)

An element \(x\) of a ring \(R\) is called nilpotent if some power of \(x\) is zero. Prove that if \(x\) is nilpotent, then \(1-x\) is a unit.

Hint: Use the geometric series.

(An ideal that is not finitely generated)

Let \(A\) be the ring of continuous real-valued functions on \(\mathbf{R}\). Let \(I \subset A\) be the set \[I = \{f \in A \mid \text{there exists } \epsilon > 0 \text{ such that } f(x) = 0 \text{ for all } x \in [-\epsilon, \epsilon]\}.\]

1. Prove that \(I \subset A\) is an ideal.
2. Prove that \(I \neq (f_1,\dots, f_{n})A\) for any \(n\) and any \(f_1, \dots, f_n \in A\).

Remark: We say that a ring is Noetherian if all its ideals are finitely generated. For example, \(\mathbf{Z}\) and all fields are Noetherian. Any quotient of a Noetherian ring is Noetherian, and a polynomial ring (in finitely many variables) over a Noetherian ring is Noetherian. As a result, many of the rings we encounter in algebra are Noetherian. The name is after Emmy Noether, who was a pioneer in modern algebra. She was the first to develop the general theory of rings and ideals.

(A generalised Chinese remainder theorem)

Let \(f, g \in R\) be two elements such that \((f,g)R = R\). Construct a ring isomorphism \[ R/fg \to R/f \times R/g.\]

Hint: The hypotheses imply that there exist \(a,b \in R\) with \(af+bg = 1\).

Remark: There is an even more general statement with ideals; see Artin, Chapter 11, Exercise 6.8.

Find all maximal ideals of \(\mathbf{Z}[x]/x^{2022}\).

Hint: First show that every maximal ideal must contain \(x\), and then use the correspondence theorem.

Remark: It is a general fact that if \(x \in R\) is nilpotent, then every maximal ideal of \(R\) contains \(x\). Your argument for \(R = \mathbf{Z}[x]/x^{2022}\) probably also proves this general statement.

Let \(F\) be a field. Read the definition of the ring \(F\llbracket t \rrbracket\) of formal power series in \(t\) with coefficients in \(F\) (Artin, Chapter 11, Exercise 2.2). Prove that \(p(t) = a_0 + a_1t + \cdots \) is a unit of \(F\llbracket t \rrbracket\) if and only if \(a_0 \neq 0\).

Find all ideals of \(F\llbracket t \rrbracket\). What are the maximal and prime ideals?

Remark: For \(F = \mathbf{R}\) or \(\mathbf{C}\), we can also consider the ring of convergent power series \(F\{ t \}\). This is a sub-ring of \(F \llbracket t \rrbracket\) consisting of power series with a positive radius of convergence. All the statements above that you proved for \(F \llbracket t \rrbracket\) remain true for \(F\{ t \}\), probably with the same proof.

Let \(F\) be a field and let \(S \subset F[t]\) be the set of polynomials \(p(t)\) such that \(p(0) \neq 0\). Prove that the homomorphism \(F[t] \to F\llbracket t \rrbracket\) extends to an injective homomorphism \[ S^{-1} F[t] \to F\llbracket t \rrbracket.\]

Remarks:

1. All the statements in Problem 1 and Problem 2 that you proved for \(F \llbracket t \rrbracket\) remain true also for the sub-ring \(S^{-1}F[t]\).
2. The map \(S^{-1} F[t] \to F \llbracket t \rrbracket\) is far from surjective. Can you write down an element that is not in the image? (Do not turn in).
3. The ring \(\mathbf{R}[t]\) is the ring of polynomial functions on the line. The rings \(S^{-1} \mathbf{R}[t] \subset \mathbf{R}\{t\} \subset \mathbf{R}\llbracket t \rrbracket\) encode increasingly permissive notions of ’functions defined around 0’.

Let \(M \subset \mathbf{R}^{n}\) be the set of vectors \(v\) such that \[ 2v \in \mathbf{Z}^n \text{ and } \sum_{i = 1}^n v_i \in \mathbf{Z}.\] Find an isomorphism of \(\mathbf{Z}\)-modules \(\mathbf{Z}^n \to M\).

An \(R\)-module is cyclic if it can be generated by one element. Prove that every cyclic module is isomorphic to the \(R\)-module \(R/I\) for some ideal \(I \subset R\).

(Artin, Chapter 14, Exercise 1.4)

A module is simple if it is not zero and it has no proper submodules. Prove that every simple \(R\)-module is isomorphic to an \(R\)-module of the form \(R/m\), where \(m \subset R\) is a maximal ideal.

Let \(M = \mathbf{Z}[i] = \{a + b i \mid a, b \in \mathbf{Z}\}\). Let \(N = (1+2i) M = \{(1+2i)z \mid z \in M\}\). Then \(M\) is a \(\mathbf{Z}\)-module and \(N \subset M\) is a sub-module. Identify the quotient \(M/N\), up to isomorphism, as a direct sum of cyclic \(\mathbf{Z}\)-modules.

(Artin, Chapter 14, Exercise 7.5)

Determine the number of isomorphism classes of abelian groups of order 400.