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Abstract Algebra — Exercise Sheet 3
Homomorphisms
1. Let φ : (G, ·) → (L, ·) be a group homomorphism. Prove that φ(eG ) = eL and φ(g −1 ) = φ(g)−1
for all g ∈ G.
2.
a) Check that f : (R, +) → (R∗ , ·) given by f (x) = 10x is an injective group homomorphism.
Is it surjective?
b) Is log : ((0, ∞), ·) → (R, +) a group homomorphism? Is it an isomorphism?
3. Let (G, ·) be a group and consider a new operation ⊙ on G given by a ⊙ b = b · a. Prove that
the inverse function defines an isomorphism of groups
(G, ·) → (G, ⊙)
a → a′ .
4. Given any constant c ∈ R, prove that the multiplication map
c : (Rn , +) → (Rn , +)
(x1 , x2 , ..., xn ) → (cx1 , cx2 , ..., cxn )
is a morphism of groups.
5. For each of the following functions, check whether it is a homomorphism:
a) f : (Z, +) → (Z, +) given by f (n) = n3 .
b) g : (Z, +) → (Z3 , +) given by g(n) = n3 ( mod 3).
c) h : (Z, +) → (Zp , +) given by h(n) = np ( mod p). Here p is a prime number.
6. Cayley’s theorem. Let (G, ⋆) be a finite group with n elements, G = {g1 , g2 , ..., gn }. Prove
that there exists an injective group homomorphism (G, ·) → (Sn , ◦).
[Hint: We have shown in class that for each element a ∈ G, the function a : G → G given by
b → a ⋆ b is bijective.]
7. Consider the map f : Z3 → Z3 given by f (n) = n + 1. Define a new operation ∗ on Z3 such that
f : (Z3 , ∗) → (Z3 , ·) is a monoid isomorphism.
8. Compare the Cayley tables for the group of symmetries of a rhombus, the group of symmetries
of a rectangle, and (Z2 × Z2 , +), where + denotes the piecewise addition
(a1 , a2 ) + (b1 , b2 ) = (a1 + b1 , a2 + b2 ).
Are these groups isomorphic?
9. The symmetry group D4 of a square has eight elements. The quaternion group H = {1; i; j; k; −1; −i; −j; −k}
(with 1 being the identity element, i2 = j 2 = k2 = −1 and ij = k; jk = i; ki = j; ji = −k; kj =
−i, and ik = −j), also has eight elements. Are they isomorphic?
10. Let f : (G, ∗) → (L, ⋆) be a group homomorphism and define the kernel of the homomorphism
to be Kerf := f −1 (eL ) = {x ∈ G; f (x) = eL }.
a) Prove that Kerf is a subgroup of G.
b) Prove that f is injective ⇐⇒ Kerf = {eG }.
11. Let f : (G, ∗) → (L, ⋆) be a group homomorphism and define the image Imf := f (G) =
{f (x); x ∈ G}. Prove that Imf is a subgroup of L.
12. Let (G, ·) be a non-commutative group and let Aut(G) = {f : G → G; f group isomorphism }.
For any g ∈ G, consider the function
ϕg : (G, ·) → (G, ·)
x → gxg−1 .
This map is called conjugation by g.
a) Prove that (Aut(G), ◦) is a group. It is called the group of automorphisms of G. What is
Aut(Z, +)? What is Aut(Z4 , +)?
b) Prove that ϕg ∈ Aut(G) for all g ∈ G.
c) Prove that there is a homomorphism ϕ : G → Aut(G) given by ϕ(g) = ϕg for all g ∈ G.
d) Prove that Kerϕ = {g ∈ G ; gx = xg∀x ∈ G}. This is called the centralizer of G. The
centralizer of an abelian group is the entire group.
13. Let (GL2 (R), ·) be the group of 2 × 2 matrices with non-zero determinant (see Q13, Ex. Set 4).
Prove that the function
f : (C∗ , ·) → (GL2 (R), ·) given by
x y
x + iy →
−y x
is an injective group homomorphism.
14. Prove that any cyclic group is isomorphic to either Z or Zn for some n ∈ N. As a corollary,
prove that all the subgroups of a cyclic group are cyclic.
15. Let X = {(12)(34), (13)(24), (23)(14)} denote the set of three pairs of opposite sides in a tetrahedron (triangular pyramid). Each element σ ∈ S4 defines a permutation
σ:X → X
(ij)(kl) → (σ(i)σ(j))(σ(k)σ(l)).
Use this to construct a map F : S4 → S3 . Find KerF .
1
b
4
b
2
b
3
b
Note: Alternatively, you can consider X = {(12)(34), (13)(24), (23)(14)} ⊂ S4 . For every σ ∈ S4 ,
prove that the map ϕσ : S4 → S4 given by ϕσ (x) = σxσ −1 satisfies ϕσ (X) = X. Hence, ϕσ
defines a bijective map from X to X. Check that this is the same as the map F : S4 → S3 . (See
Q11 in Ex.Set3).
16. Consider the operations ⊞ and ⊠ : Z6 × Z6 → Z6 given by
a ⊞ b := a + b + 1,
a ⊠ b := 5ab + 5(a + b) + 4.
Prove that (Z6 , ⊞) and (Z6 , ⊠) are monoids. What is the identity element in each case? Is ⊠
distributive with respect to ⊞?
Let + and × be the usual addition and multiplication on Z6 . Find c ∈ Z6 such that f (n) = 5n+c
defines an isomorphism from (Z6 , +) to (Z6 , ⊞) and from (Z6 , ×) to (Z6 , ⊠).
17. Let (Mn×n (R), ·) denote the set of n × n matrices with real entries. Given A ∈ Mn×n (R), we
denote by Aij the entry on the i-th row and j-th column of A. We define the product AB by
(AB)ij =
n
X
Aik Bkj = Ai1 B1j + Ai2 B2j + · · · Ain B2n .
k=1
For any matrix A ∈ Mn×n (R), let At denote the transpose matrix, obtained from A by swapping
rows with columns Prove that the transpose function defines an isomorphism of monoids
(Mn×n (R), ·) → (Mn×n (R), ⊙)
A → At .
where by definition A ⊙ B = BA.
[Hint: See solutions of Q14 in Ex. Set 4]
18. Prove that the determinant function A → detA is a group homomorphism
det : (GL2 (R), ·) → (R∗ , ·).
19.
a) Prove that
f : (Sn , ◦) → (GLn (R), ·) given by
α → f (α), where f (α)ij =
1 if i = α(j),
0 otherwise ,
is an injective homomorphism.
b) If n = 3 and α = (132), check that det(f (α)) = sign (α).
20. Consider the function f : Z → Z13 × Z11 given by f (a) = (a( mod 13), a( mod 11)).
• Check that f : (Z, +) → (Z13 × Z11 , +) is a group homomorphism.
• Prove that Ker(f ) = 143Z.
• Prove that ([1], [0]) and ([0], [1]) are in the image of f . Hence or otherwise, prove that f is
surjective and (Z143 , +) ∼
= (Z13 × Z11 , +).
• Generalize the result above by replacing 13 and 11 with any two relatively prime numbers.
Prove your results.