Mathematics 676
Homework (due Apr 7)
A. Hulpke
42) Let G = ⟨a, b, c ∣ a 2 = b 3 = c 4 = (ab)2 = (ac)2 = (bc)3 = 1, [c 2 2, b]⟩. (Note: [x, y] = x − 1y − 1x y
is called the commutator od x and y.)
Let S = ⟨a, c⟩ ≤ G. By enumerating the cosets of S in G, determine permutations for the action of G
on the cosets of S.
43) Let G = ⟨a, b, c ∣ b a = b 2 , c b = c 2 , a c = a 2 ⟩ and S = ⟨a, b a ⟩. Perform by hand a coset enumeration for S as a subgroup of G. Use the result to conclude that G must be trivial.
44) Let H = ⟨a, b ∣ a 2 = b 3 = (ab)7 ⟩ (this group is called the Hurwitz Group). Using a low index
algorithm, show that:
a) H has no proper subgroups if index < 7.
b) PSL2 (7) can be generated by two elements x, y with ∣x∣ = 2, ∣y∣ = 3, ∣x y∣ = 7.
c) n = 15 is the smallest value of n such that A n can be generated by elements x, y with ∣x∣ = 2, ∣y∣ = 3,
∣x y∣ = 7.
45) Let G = ⟨g ∣ R⟩ be a finitely presented group. Show that by adding all commutators [g i , g j ] for
g i , g j ∈ g to R, we obtain a presentation for G/G ′ .
46) Let G = ⟨a, b ∣ aba 2 b −1 a, ab2 a −1 b 2 ⟩ and S = ⟨a, a b , bab −1 ⟩ ≤ G.
a) Show that [G ∶ S] = 4 and determine a coset table.
b) Define Schreier generators for S, and – using an augmented coset table – write a presentation for
S in the Schreier generators.
47) Let G = ⟨a, b ∣ aba −2 bab −1 , (b −1 a3 b−1 a −3 )2 a⟩. Using the Low-Index algorithm as implemented
in GAP, find a subgroup S ≤ G such that [S ∶ S ′ ] is infinite (test AbelianInvariants for S), and
thus G is infinite.