HOMEWORK 3, MATH 60210, BASIC ALGEBRA, DUE TUESDAY, SEPT.
15
INSTRUCTOR, SAM EVENS, FALL 2009
INSTRUCTIONS: Do 7 of these 13 problems.
1. Let Xbe a G-set, where Gis a group. Define a map φ:G→AXby φ(g) = lg, where
lg(x) = g·xfor x∈X. Prove that φis a group homomorphism, and Ker(φ) = ∩x∈XGx.
2. For a group G, let Int(G) = {cx:x∈G}, where cx(g) = xgx−1. Let Aut(G) is the set of all
isomorphisms φ:G→G.
(a) Prove that Aut(G) is a subgroup of AG.
(b) Prove that Int(G) is a normal subgroup of Aut(G), and Int(G)∼
=G/Z(G).
3. Ash, 1.4, problem 6.
4. (see D+F, problem 18 of 3.2): Let Gbe a finite group and let Hand Nbe subgroups of G
with Nnormal in G. Prove that if |H|and [G:N] are relatively prime, then H⊂N.
5. Prove that R/Z∼
=S1, where S1={z∈C∗:|z|= 1}.
6. Prove that C/Z∼
=C∗.
7. Let Fbe a field. Prove that T(n, F )/U(n, F )∼
=D(n, F ).
8.Let kand nbe positive integers, and consider the group homomorphism π:Z→Z/nZ. Show
π(kZ) = π(dkZ) for some dkdividing n, and compute dkin terms of nand k.
9. Recall the dihedral group G=D2nintroduced in lecture, or as presented in Ash, 1.2.3, so
|D2n|= 2n. Compute Z(G) (it depends on n).
10. Compute the group Int(D2n).
11. Let G=D2n, so Gis the group generated by a rotation r=r2π/n and a reflection s, where
s(x, y) = (x, −y). Let kbe a divisor of n, and let Hk:= {rjk :j= 0,..., n
k}. Prove that Hkis
a normal subgroup of G.
12. (notation from previous problem) Is G/Hka dihedral group D2m? If so, which one? Prove
that your answer is correct.
13. (cf. D+F, problem 6) Show that R∗/{±1}∼
=R>0, the group of positive real numbers under
multiplication.
