Math 911 - Section 1 - Spring 2012

Math 911 - Section 1 - Spring 2012 - Homework Page

Chapter 1:

(Ch1.1) For the group G:= < a,b | a² = b³ = 1, aba^-1 = b² >, show that G is isomorphic to a symmetric group S_n for some n (which you will need to determine), and find the shortlex normal forms for this finitely presented group.
(Ch1.2) Find (and prove) a sequence of Tietze transformations showing that the presentations < a, b | aba = bab > and < x, y | x² = y³ > present isomorphic groups. (This is the braid group on 3 strands.)
(Ch1.3) For each of the groups below, find a set of normal forms and draw the Cayley graph (and Cayley complex) with respect to the given presentation.

F₂ × Z = < x, y, z | [x,z] = 1, [y,z] = 1 >
Z² ∗ Z = < x, y, z | [x,y] = 1 >

(Ch1.4) (*¹) For each of the groups below, find a set of normal forms and draw the Cayley graph with respect to the given presentation.

G₁ = < x, y, z | [x,y] = 1, zxz^-1 = xy², zyz^-1 = y >
G₂ = < x, y, z | [x,y] = 1, zxz^-1 = x^-1, zyz^-1 = y^-1 >
G₃ = < x, y, z | [x,y] = 1, zxz^-1 = x², zyz^-1 = y >

(Ch1.5) Prove the Induced Homomorphism Theorem: If h : G -> H is a group homomorphism and N is a normal subgroup of G contained in the kernel of h, then there is a unique homomorphism h' : G/N -> H satisfying h' q = h, where q : G -> G/N is the quotient map.
(Ch1.6) (*¹) For the group G = Z₄ × Z₆, compute the groups Inn(G), Aut(G), and Out(G).
(Ch1.7) (*¹) Prove that a group G is a semidirect product N ⋊_α H of two groups N and H iff G is a split extension of N by H. Also determine which of the groups G₁, G₂, G₃ in Problem (Ch1.4) is/are a semidirect product of < x,y > by < z >.
(Ch1.8) For each of the following classes (sets) of groups, determine whether the class is closed under taking direct products, and/or closed under taking semidirect products.

Infinite groups
Free groups
Right-angled Artin groups

Chapter 2:

(Ch2.1) Show that if 1 = G₀ ◁ G₁ ◁ ... ◁ G_n = G is a central-normal series for a group G, then γ_n-i+1(G) ≤ G_i ≤ ζ_i(G) for all 0 ≤ i ≤ n.
(Ch2.2) Prove that the class of nilpotent groups is closed under

subgroups,
quotients, and
(*²) finite direct products.

(Ch2.3) (*²) Prove the following portions of the theorems from class on HNN extensions:

Show that for all integers m,n,p,q, the group G = < x,y,z | [x,y] = 1, zxz^-1 = x^myⁿ zyz^-1 = x^py^q > is solvable.
For each of the groups G₁, G₂, G₃ in Problem (Ch1.4), determine whether or not the group is also nilpotent or polycyclic. (Note: You must prove your answer directly, and may not use the theorems from class on ascending HNN extensions.)

(Ch2.4) Suppose that G is a finitely generated group, and for each natural number i let γ_i be the ith group in the lower central series for G.

Show that for all natural numbers n and for all 0 ≤ i ≤ n the subgroup γ_i/γ_n of G/γ_n is finitely generated. (Hint: Use the commutator identities [a,bc]=[a,b][a,c][[c,a],b] and [ab,c]=[b,c][[c,b],a][a,c] (or any other commutator identities you might find!).)
Show that if G is a finitely generated nilpotent group, then γ_i is a finitely generated subgroup for all i.
Compute the lower central series for the Heisenberg group.

(Ch2.5) A group G satisfies the property min if G does not have any strictly descending chain G > G₁ > G₂ > ... of subgroups of G (where G_i ≠ G_i+1 for every i). Determine whether or not the class of min groups is closed with respect to subgroups, quotients, and/or extensions.
(Ch2.6) A group G is supersolvable if G is poly-normal-cyclic; that is, if there is a sequence 1 = G₀ ◁ G₁ ◁ ... ◁ G_n = G in which each quotient G_i+1 / G_i is cyclic and each group G_i is normal in G. [This rather unfortunate name is used throughout the literature!]

(*² extra credit) Either show that the class of supersolvable groups is closed under taking subgroups, or show that the class of supersolvable groups is not closed under taking extensions.
(Hint if you prefer to try the extension counterexample: Following Robinson's problem 6 on p. 157, let X = < a,b | a² = b² = (ab)⁴ = 1 > = D₈, A = < c,d | cd = dc, c³ = d³ = 1 > = Z₃ × Z₃, and let φ : X → Aut(A) be the homomorphism defined by φ(a)(c) = d, φ(a)(d) = c, φ(b)(c)=c^-1, and φ(b)(d) = d. Let G be the semidirect product A ⋊_φ X. Show that A and X are supersolvable. To show that G is not supersolvable, since G is finite, there are only finitely many possible cyclic series to consider.)
(*²) Find (and prove!) an example of a finitely generated group that is supersolvable but not nilpotent.

(Ch2.7) Let P be an abstract property of groups (i.e. a property of groups invariant under isomorphism, not a property of subgroups [eg finite, abelian, nilpotent, solvable, but not normal, central].)

Show that if P is preserved when taking extensions, then poly-P = P.
Show that if P is inherited by subgroups, then every subgroup of a poly-P group is poly-P.
(*²) Show that if P is closed under taking quotients, then every quotient of a poly-P group is poly-P.

(Ch2.8) Let P be an abstract property of groups.

Show that if P is a property inherited by subgroups, then
- every subgroup of a residually-P group is residually-P, and
- a group G is locally-P if and only if every finitely generated subgroup of G is P.
Show that if P is a property perserved by taking quotients, then every quotient of a locally-P group is locally-P.

(Ch2.9) For each of the following, prove or disprove that the property is preserved by taking direct products:

Polycyclic
Virtually abelian
Residually finite
Locally finite

Chapter 3:

(Ch3.1) Show that quasi-isometry is an equivalence relation.
(Ch3.2) Two groups G and H are quasi-commensurable if there exist subgroups M ◁ K ≤ G and N ◁ L ≤ H such that |G:K|, |H:L|, |M|, and |N| are all finite and K/M ≅ L/N.

Show that quasi-commensurability is an equivalence relation on finitely generated groups
(*³) Show that commensurability ⇒ quasi-commensurability ⇒ quasi-isometry for finitely presented groups.

(Ch3.3) Draw the Cayley graph for the group G = < a, b, c | a²=b²=c²=1, (ab)²=(ac)⁴ = 1 >. Then explain how you can see from this Cayley graph that G is quasi-isometric to the free group on two generators.
(Ch3.4) (*³) Suppose that G is a finitely generated group with (ball) growth function β_G.

Show that if H ≤ G is a finitely generated subgroup, then β_H ≤ β_G.
Show that if N ◁ G, then β_G/N ≤ β_G.

(In both parts of this problem, ≤ means the ``curly'' ≤ condition. Eg in part 1: There is a constant λ such that β_H(n) ≤ λ β_G (λ n + λ) + λ for all n in N.)
(Ch3.5) Let G = < a, b, c | a²=b²=c²=1, (ab)³=(ac)³=(bc)³ = 1 >. The Cayley graph for this Coxeter group G with respect to the generating set A = {a,b,c} embeds in the plane as the triangulation by regular hexagons. Compute the spherical and ball growth functions and growth series for G with respect to A.
(Ch3.6) Find a formula for the spherical growth series of a free product G₁ * G₂ * G₃ of any 3 groups in terms of the spherical growth series of the groups G_i.
(Ch3.7) Show that a finitely generated group G has a linear (ball) growth function (that is, β_G ≈ (n → n)) if and only if G is quasi-isometric to Z.
(Ch3.8) Let G be the Coxeter group G = < a, b, c | a²=b²=c²=1, (ab)³=(ac)² = 1 >.

Find a shortlex complete rewriting system for G.
Draw a FSA (finite state automaton) whose language is the set of shortlex normal forms for G.
Compute (using the matrix method) the spherical and ball growth series for G with respect to the generating set {a, b, c}, as rational functions.

(Ch3.9) (*³) Let G = < a, b, c | ab = ba, c² = 1, cac = b >; that is, G is a semidirect product of Z² with Z₂. Let A := {a^±1, b^±1, c}.

(a) Find a shortlex complete rewriting system for G over A with a < a^-1 < b < b^-1 < c.
(b) Draw a FSA (finite state automaton) whose language is the set of shortlex normal forms for G (with respect to the ordering in (a)), and write a regular expression for this language.
(c) Draw the Cayley graph of G over A.
(d) Compute, using the matrix method, expressions for the spherical and ball growth series of (G, A) as rational functions.
(e) Find formulas for the spherical and ball growth functions.
(You can either use your rational functions in (d) or justify your formulas using your Cayley graph from (c).)
(f) Find a regular expression for the language of geodesics of G over A.
(Justify using your Cayley graph from (c).)
(g) Find a formula for the strict geodesic growth function for G with respect to A.
(Here again you have various choices: Either justify your formula by finding an FSA for your geodesic language in (f) and then using the matrix method to find the series and then the function, or by directly counting numbers of words of various lengths from your regular expression in (f), or by counting paths using your Cayley graph from (c).)

Problems marked with (*¹) are due to be handed in at the start of class on Thursday, February 2.
Problems and parts of problems marked with (*²) are due to be handed in at the start of class on Thursday, March 1.
Problems and parts of problems marked with (*³) are due to be handed in at the start of class on Thursday, April 19.

S. Hermiller.