HOMEWORK 2, MATH 60210, BASIC ALGEBRA, DUE TUESDAY, SEPT. 8
INSTRUCTOR, SAM EVENS, FALL 2009
INSTRUCTIONS: Do 7 of these 12 problems.
1. Let Xbe a set and suppose |X| ≥ 3. Show that AX, the set of bijections of X, is a nonabelian
group (cf. Ash, 1.2, problem 5).
2. Ash, 1.3, problem 4.
3. Ash, 1.3, problem 6.
4. Ash, 1.3, problem 8.
5. Ash, 1.3, problem 9.
6. Let Gbe a group and let Hand Kbe two subgroups of Gof finite index, i.e., [G:H] and
[G:K] are finite. Prove that H∩Kis a subgroup of finite index in G.
7. If xis an element of a group G, we say xis a torsion element if the order |x|is finite.
(a) If Gis abelian, show the set Gtor of all torsion elements of Gis a subgroup.
(b) Give an example of a nonabelian group where Gtor is not a subgroup.
8. (cf. Dummit and Foote, problem 14 of 3.1) By definition, a group Gis divisible if for each
integer nand each a∈G, there exists b∈Gsuch that bn=a.
(a) Show that every element of G=Q/Zhas finite order, but for each positive integer m, there
exists an element of Gof order m.
(b) Show that Q/Zis the torsion subgroup of R/Z.
(c)(extra credit) Construct an isomorphism from Q/Zto the subgroup of torsion elements of
C∗. Here C∗is the nonzero complex numbers, which is a group under multiplication.
9. (cf. D+F, problem 15, 3.1) By definition, a group Gis divisible if for each integer nand
each a∈G, there exists b∈Gsuch that bn=a. Show that if Gis divisible, and Nis a normal
subgroup of G, then G/N is divisible. Show that Qand Q/Zare divisible.
10. For a group Gwith subgroup H, let the normalizer NG(H) = {g∈G:gHg−1=H}. If H
is finite, prove that NG(H) = {g∈G:gHg−1⊂H}.
11. (D+F, 3.1, problem 36) For a group Gwith center Z(G), prove that if G/Z(G) is cyclic,
then Gis abelian.
12. (D+F, 3.2, problem 4) Let Gbe a group of order pq, where pand qare prime. Prove that
either Gis abelian or Z(G) = {e}.
