Mathematics 366
Homework (due Dec. 9)
72) Let G = Sn and ai = (i, i + 1) for i = 1, . . . , n − 1.
a) Let g ∈ Sn with and ng = i. Show that
A. Hulpke
a1: Move left string over middle one,
a2: Move middle string over right one,
b = g · ai · ai+1 · · · · · an−1
a1
a2
a1
a2
fulfills that nb = n, i.e. that b ∈ Sn−1 .
a1
a2
b) Using the argument from a), show by induction over n
that Sn = ha1 , . . . , an−1 i.
c) Show that ai a j = a j ai if |i − j| ≥ 2 and ai a j ai = a j ai a j if |i − j| = 1. (Note: These relations are
called “braid relations”, the name comes from representing the braiding of strings. As shown in
the picture on the side, relations of this type hold for the “twists” of adjacent strings.
One can show that using these braid relations as well as relations a2i = e, one can bring every word
in Sn into a shortest form.)
73) Let G be a group and H, K ≤ G with H ≤ K ≤ G. Show that if [G : H] is finite, then [G : H] =
[G : K] · [K : H].
74∗ ) Assume that G, H and K are as given in problem 73.
Let {ri } a set of representatives of the cosets of K in G and {s j } a set of representatives of the
cosets of H in K. Show that the products {s j · ri } form a set of representatives for the cosets of H
in G. (In other words: Show that for every x ∈ G there is a pair r, s such that Hx = H(s j ri ), and that
H(s j ri ) = H(sy rx ) only if i = x and j = y.)
75) Describe (e.g. find a set of representatives) the cosets of Sn−1 in Sn .
76∗ )
For a group G, define the derived subgroup as
G0 := x−1 y−1 xy | x, y ∈ G ,
thus every element of G0 is a product of elements of the form xi−1 y−1
i xi yi .
0
a) Prove that G C G.
b) Prove that G/G0 is abelian.
c) If N C G with G/N abelian, prove that G0 ≤ N.
77) Let G ≤ Sn be a permutation group and Ω = {(a, b) | 1 ≤ a, b, ≤ n} the set of all pairs of
points from {1, . . . n}. (Sometimes this set is written as Ω = {1, . . . n} × {1, . . . n} and called the
cartesian product.) We know that |Ω| = n2 . We define an action α of G on Ω by acting on the
entries:
α ({a, b}, g) := {ag , bg }
a) Show that this defines a group action.
b) For G = D8 = h(1, 2, 3, 4), (1, 3)i, determine the orbits of G on Ω.
78) A dodecahedron is a regular solid with 12
faces, each of which is a regular pentagon. At each
corner three pentagons meet. We consider rotational
symmetries.
a) How long is the orbit of one face?
b) How large is the stabilizer of one face?
c) Determine the number of rotational symmetries of
a dodecahedron.
79) a) Prove, that a factor group of a cyclic group is cyclic.
b) Let G be a group and N C G with [G : N] = m. Show that |x| is a divisor of m for every x ∈ G/N.
80) Which of the following functions ϕ: G → H are group homomorphisms? Which are isomorphisms? Prove your statements!
a) G = H = Z8 . ϕ1 : G → H, x 7→ 3x mod 8.
b) G = H = Z8 . ϕ2 : G → H, x 7→ 4x mod 8.
c) G = H = Z8 . ϕ3 : G → H, x 7→ x−1 mod 8.
d) G = H = D8 . ϕ4 : G → H, x 7→ x−1 .
e) G = H = Z10 . ϕ5 : G → H, x 7→ x + 2 mod 10.
f) G = H = U(5). ϕ6 : G → H, x 7→ 2x mod 5.
g) G = GL2 (Q), H = (R, ×). ϕ7 : G → H, x 7→ det(x).
h) G = Z5 , H = (Z, +). ϕ8 : G → H, x 7→ x.
81∗ ) Prove or disprove:
a) U(20) and U(24) are isomorphic.
b) U(20) and D4 are isomorphic.
c) U(20) and U(15) are isomorphic.
82∗ ) Let ϕ: G → H be a homomorphism and a ∈ G.
a) Show that ϕ(CG (a)) ⊆ CH (ϕ(a)).
b) Show that if ϕ is an isomorphism, then ϕ(CG (a)) = CH (ϕ(a)).
83∗ ) Let G be a group of order 4. Show that G is isomorphic either to Z4 or to the group
h(1, 2)(3, 4), (1, 3)(2, 4)i.
Hint: If G is not cyclic, show that all elements of G must have order 1 or 2.
Problems marked with a ∗ are bonus problems for extra credit.