Actions of $S_n$.

• Lemma 4.6 of the Permutation Groups paper by Borovik and Cherlin
• Borovik-Nesin: Section 5.5.

• Focus on working out the details of Lemma 4.6 (and finding the error(s)).
• (In case Lemma 4.6 becomes overwhelming...) Borovik-Nesin: pp. 93 #8, 10, 11 (both 10 and 11 have hints in the back)

Time to breathe.

• Borovik-Nesin:
  1. Browse Section 5.4 (time permitting). Corollary 5.32 is a highlight as it says that a subgroup, with no right to be definable, is definable.

• Borovik-Nesin: pp. 78-79 #13

Connected Component.

• Borovik-Nesin:
  1. Macintyre' Theorem: p. 72 Exercise #9
  2. Section 5.2 (Lemma 5.9 is incredible important, as are many of the other results)

• Borovik-Nesin: pp. 72-73 #13(a,b),14
  • The back of the book (p. 380) has hints for both of these problems.
• Borovik-Nesin: pp. 78 #3, 4, 8, 9
  • The back of the book (p. 380) has hints for several of these too.
  • For #3, first prove the following result: if $G$ is any group with a subgroup $H$ of finite index, then $G$ has a normal subgroup of finite index, which is contained in $H$. Hint: Let $G$ act on the (finite) set of left cosets of $H$ in $G$. This gives a homomorphism of $G$ into $S_n$ where $n$ is the index of $H$ in $G$. What can you say about the kernel?

Descending Chain Condition.

• Borovik-Nesin:
  1. Chapter 5 pp. 68-70 (aim to understand all proofs)
  2. Section 1.3 (and necessary definitions on p. 3)
     • This is to acquaint you a bit with the notion of divisibility before working on the problems.
     • Skim the exercises from Section 1.3. (Try to prove anything that appeals to you.)

• Revisit and tidy up Borovik-Nesin: p. 27 #6 (skip the "and that then..." part)
• Borovik-Nesin: p. 71 #2, #7 (but not the part about $\operatorname{PSL}_2(K)$), #10, #11
  • Here's a hint for #10: since $A$ is abelian, the map $A\rightarrow A: a \mapsto a^n$ is a homomorphism. Notice that you are trying to show that this map is surjective. What do you know about the kernel? What can you conclude about the rank and degree of the image? (Look back at Lemma 4.17.) Tie it all together with Lemma 5.1.
  • Here's a hint for #11: the idea is to show that every nontrivial $g$ is contained in a definable and abelian subgroup (which inherits the same property about its elements); then you can apply #10. Think centralizers and centers...

Morley rank.

• Borovik-Nesin:
  1. Chapter 2 (This is mostly a review, so feel free to just skim.)
  2. Section 3.1 (starting on page 30)
  3. Chapter 4:
     • Skim Section 4.1.1 (it is simply abstracting the closure properties we know for definable, and interpretable, sets)
     • Section 4.1.2 (this is an abstract version of Morley rank---it coincides with Morley rank for $\omega$-stable groups, but not necessarily for other structures)
     • Section 4.2 (most of the proofs are good ones to work through)

• Revisit and tidy up the problem from last week. (The outline has been updated.)
• Borovik-Nesin: p. 27 #6 (skip the "and that then..." part)
• Borovik-Nesin: p. 32 #1
• Borovik-Nesin: pp. 64-65 #9 (hint: lemma 4.22), #12, #13

Saturation.

• Marker: Section 4.3
  1. Beginning through proof of Corollary 4.3.5. (Aim to understand all proofs.)
  2. Example 4.3.9
  3. Statement of Theorem 4.3.12 and 4.3.15
  4. Definition 4.3.16 through Corollary 4.3.19

• Definition: A structure $M$ is minimal (with respect to some language extending the group language) if every definable subset of $M$, using parameters from $M$, is either finite or cofinite.

  Problem: Prove that every infinite, minimal group is abelian.

  Here are some possible steps to follow:

  1. Show that if $G$ is minimal, then every proper definable subgroup of $G$ is finite.
     Hint: let $H$ be a proper definable subgroup of $G$ and consider the cosets of $H$.
  2. Show that if $G$ is infinite and minimal, then for all $x,y \in G - Z(G)$, $x$ and $y$ lie in the same conjugacy class, i.e. there is at most one noncentral conjugacy class. Also show that every $x\in G - Z(G)$ has finite order.
     Hint: let $x,y\in G - Z(G)$, and show that $x^G$ and $y^G$ (the conjugacy classes of $x$ and $y$) are both infinite by considering their relationship to $C_G(x)$ and $C_G(y)$. Use this to show that $x^G$ and $y^G$ must intersect nontrivially.
  3. Conclude from the previous part that if $G$ is infinite and minimal then all nonidentity elements of the quotient $G/Z(G)$ are conjugate and of finite order.
  4. Show that in any group $H$ for which all nonidentity elements have order $p$ for some prime $p \ge 3$, if $1\neq a\in G$, then $a$ is not conjugate to $a^{-1}$. Conclude that in any such group $H$, $H$ must contain at least $3$ (distinct) conjugacy classes.
     Hint: assume that $a^h = h^{-1}ah = a^{-1}$. Now compute $a^{h^p} = h^{-1}\cdots h^{-1}h^{-1}ahh\cdots h$ two different ways; note that $(a^{-1})^h = (a^h)^{-1}$ (do you see why?).
  5. Show that if $H$ is any group for which all nonidentity elements are conjugate and of finite order, then $|H| = 2$.
     Hint: show, in sequence, that (1) all nonidentity elements have the same prime order $p$, (2) $p$ must be $2$, (3) $H$ is abelian, and (4) $|H| = 2$.
  6. Tie it all together to prove that if $G$ is infinite and minimal, then $G$ is abelian.

Omitting types, prime models, and homogeneity.

• Marker: review Section 4.1 (pp. 115-118 and Example 4.1.12)
• Marker: Section 4.2
  1. Beginning through statement of Theorem 4.2.3 and statement of Theorem 4.2.4
  2. Statement and proof of Theorem 4.2.5
  3. Definition 4.2.6 and the words right before and after it.
  4. Definition 4.2.6, Definition 4.27, and statement of Theorem 4.2.8; also skim the words in between.
  5. Subsection on Countable Homogeneous Models: everything!

• Marker: 4.5.1, 4.5.2 (just part about $(\mathbb{Z},s)$), focus on working through proofs in section on Countable Homogeneous Models
• Marker: Write up 3.4.33(a)(b) from last time.

More with orderings and on to types.

• Marker: Review Chapter 3 topics from last time; Section 4.1 (just pp. 115-118 and, time permitting, Example 4.1.12)

• Marker: Problems 3.4.1, 3.4.9, 3.4.33(a)(b) (see definition 3.1.18), 4.5.1

Quantifier elimination and more.

• Marker: Sections 3.1 (fairly thoroughly through p. 79, then skim the rest), 3.2 (just up to, but not including, Zariski Closed and Constructible Sets), 3.3 (skim up to, but not including, Semialgebraic Sets)

• Marker: Problems 3.4.1, 3.4.3 (you can skip the $\operatorname{acl}(A)$ part), 3.4.4, 3.4.33(a)(b) (see definition 3.1.18)

Up and down, back and forth.

• McNulty's book: pp. 40-49
• Marker: Sections 2.2, 2.3, beginning of 2.4 (this should mostly overlap and reinforce the McNulty reading)

• Marker: Problems 2.5.1, 2.5.3, 2.5.6 (for meditation purposes), 2.5.14, 2.5.28 (a)(b)

Next stop, ultrafilters.

• McNulty's book: pp. 32-38 (skim 36-37, but be on the lookout for where properties of ultrafilters come into play)

• McNulty's book: Problems 12 (on p. 31) and 14, 15, 16, 17 (on p. 39)
• Tent-Ziegler Book: Exercises 2.2.3, 2.2.4

On to compactness.

• Marker: review key concepts from Ch. 1
• Marker: all of §2.1, but just skim (or skip for now) the proof of Lemma 2.1.7
• McNulty's book: pp. 21-28 (skim §2.1, but pay close attention to the applications of compactness in §2.2)

• Marker: 1.4.3, 1.4.11
• McNulty's book: Problems 9, 10, 11 (on p. 31)

Getting started on Chapter 1 of Marker's book.

• §1.1: all
• §1.2: all
• §1.3: definability, first few examples, example at bottom of page 20, 1.3.5, 1.3.6