Mathematics 676 Homework (due Apr 14) 46) A. Hulpke

Mathematics 676
Homework (due Apr 14)
A. Hulpke
46) The Fibonacci group Fib n is defined as
Fib n = ⟨g0 , . . . , g n−1 ∣ g i g(i+1 mod n) = g(i+2) mod n , i = 0, . . . , n − 1⟩ .
For n ≤ 10 determine for G = Fib n the structure of G/G ′ and G ′ /G ′′ . Can you determine finiteness
of any of these groups?
47) Show that the free group of rank 2 contains for every n ≥ 2 a normal subgroup that is free of
rank n.
48) Let G be a finitely presented group given by a presentation with fewer relators than generators.
Show that G is infinite. (Hint: Consider G/G ′ )
49) Using a composition series, compute a presentation for the solvable group GL(2,3). (You may
use GAP for calculations short of having it compute a presentation.)
50) Show that for every prime p > 2 there are exactly two nonisomorphic nonabelian groups of
order p3 , namely the group ⟨x, y ∣ x p = y p = 1, x y = x p+1 ⟩ (which has a cyclic normal subgroup of
order p2 ) and ⟨x, y, z ∣ x p = y p = z p = 1, y x = yz, z x = z y = 1⟩ (in which all elements have order 1 or
p, but which is not C 3p ).
Hint: You know from M566 (G/Z(G) cyclic implies G abelian) that ∣Z(G)∣ = p and G/Z(G) ≅ C 2p .
51) For a finite group G we define the Frattini subgroup Φ(G) as the intersection of all maximal
subgroups of G.
a) Show: Φ(G) consists of those elements of G which are redundant in every generating system of
G. (I.e. Φ(G) = {x ∈ G ∣ ∀S ⊂ G ∶ (G = ⟨S ∪ {x}⟩ ⇒ G = ⟨S⟩)}.)
b) Now consider that G is a p-group (i.e. ∣G∣ = p a for a prime p). Show that
Φ(G) = G ′ G p = {g p , [g, h] ∣ g, h ∈ G}
(i.e. it is the smallest normal subgroup N ⊲ G, such that G/N ≅ C pk .)
(Hint: Homework problem 14 shows that every maximal subgroup of a p-group has index p.)
c) (Burnside’s Basis Theorem) Let G be a p-group and [G ∶ Φ(G)] = p k . Show that every irredundant generating set (i.e. no element can be removed without generating a smaller subgroup) of G
contains k elements. (We call k the rank of the p-group.)