Name:
Problem Set 5
Math 415 Honors, Fall 2014
Due: Tuesday, October 21.
Review Sections 11, 13 in your textbook.
Complete the following items, staple this page to the front of your work, and turn your assignment
in at the beginning of class on Tuesday, October 21. Remember to fully justify all your answers,
and provide complete details. Neatness is greatly appreciated.
1. Find all the subgroups of Z3 × Z3 × Z4 .
2. Is Z5 × Z9 isomorphic to Z15 × Z3 ? Is Z15 × Z6 × Z7 isomorphic to Z10 × Z3 × Z21 ? Justify your
answers.
3. Let G1 , . . . , Gn be abelian groups. Prove that G1 × · · · × Gn is abelian.
4. Let G1 , . . . , Gn be groups (not necessarily abelian), with respective identities e1 , . . . , en . Prove
that the functions φi : Gi → G1 × · · · × Gn defined by φi (x) = (e1 , . . . , ei−1 , x, ei+1 , . . . , en ) are
injective group homomorphisms.
5. Let G be an abelian group. Define H = {x ∈ G | x2 = e}. Show that H is a subgroup of G.
Give an example of a nonabelian group G where H as above is not a subgroup of G.
6. Let G be a finite abelian group, and assume that G is not cyclic. Prove that G has a subgroup
isomorphic to Z p × Z p , for some prime number p. Give an example where this statement is not
true for an infinite abelian group G.
7. Let H, K, G be finitely generated abelian groups. Suppose that G × K is isomorphic to H × K.
Prove that G H.
8. Let G be a group and a ∈ G. Define a function φ : Z → G via φ(1) = a and φ(n) = an . Prove
that φ is a homomorphism.
9. Let φ : Z → S 5 be a homomorphism defined by φ(1) = (1, 2, 5)(2, 3) (use the previous
exercise). Compute the kernel of φ.
10. Let φ12 : G1 → G2 and φ23 : G2 → G3 be group homomorphisms. Show that φ23 ◦ φ12 : G1 →
G3 is a group homomorphism.
11. Let G be a finite group, and let φ : G → G0 be a group homomorphism. Prove that the image
of φ is a finite subgroup of G0 . Prove that the order of the image of φ divides the order of G.
Show that there are no nontrivial homomorphisms φ : Z2 → Z7 .
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Through the course of this assignment, I have followed the Aggie
Code of Honor. An Aggie does not lie, cheat or steal or tolerate
those who do.
Signed:
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