ALGEBRA MIDTERM
No credit will be given for an answer without reasoning.
1.
(1) [5%] Let hG, ∗i be a group with identity element e. Suppose that a, b, c ∈ G such that a ∗ b ∗ c = e. Show that b ∗ c ∗ a = e also.
(2) [5%] Check that S3is not abelian.
2.
(1) [5%] Let Q∗ = { a ∈ Q | a 6= 0 }. Define a ∗ b = a/b for a, b ∈ Q∗. Does hQ∗, ∗i form a group? Why or why not?
(2) [5%] Let H be a subgroup of Z30generated by the element 4. Find the index (G : H).
3.
(1) [5%] Write the element Ã
1 2 3 4 5 6 7 8 9 10 5 8 6 1 3 9 10 7 2 4
!
∈ S10
into a product of disjoint cycles.
(2) [5%] Classify the group (Z × Z)/h(1, 3)i according the fundamental theorem of finitely gener- ated abelian groups.
4.
(1) [5%] Is there a nontrivial homomorphism φ : Z4 → Z? Why or why not?
(2) [5%] Is there a nontrivial homomorphism φ : Z4× 2Z → Z × Z? Why or why not?
5. Let GL(n, R) be the general linear group of degree n.
(1) [5%] Check that GL(n, R) is a group under the matrix multiplication.
(2) [5%] Define O(n, R) = { A ∈ GL(n, R) | AAt= I } where Atdenotes the transpose of A and I denotes the identity matrix. Show that O(n, R) is a subgroup of GL(n, R).
6. Let φ : Z24→ S8be a homomorphism given by φ(1) = (13)(24)(517).
(1) [5%] Find φ(14).
(2) [5%] Find the kernel of φ.
7. [10%] Let X = { (a, b) ∈ Z × Z | b 6= 0 }, and define a relation “∼” on X by (a, b) ∼ (c, d) if ad = bc.
Show that “∼” is an equivalence relation.
8. [10%] Show that U × R+' C∗as groups where C∗ = { z ∈ C | z 6= 0 }, U = { z ∈ C | |z| = 1 } and R+= { r ∈ R | r > 0 }.
1
2 ALGEBRA MIDTERM
9. [10%] Let φ : G → G0 be a group homomorphism. Show that φ[G] is abelian if and only if for all x, y ∈ G, we have xyx−1y−1∈ Ker(φ).
10. Let G be a group and let Aut(G) be the set of all automorphisms of G. It is known that Aut(G) forms a group under function composition. (You don’t need to check this fact.) Let H be the set of all inner automorphisms of G.
(1) [5%] Show that H is a subgroup of Aut(G).
(2) [5%] Show that H is a normal subgroup of Aut(G)
