This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 4124 Wednesday, February 1 Second Homework Solutions 1. 2.1.10(a) on page 48 Prove that if H and K are subgroups of the group G , then so is H K . Let e be the identity of G . Then e H , K because H and K are subgroups of G , consequently e H K . Next let x , y H K . Then xy H because H G and xy K because K G . Therefore xy H K . Finally x 1 H and K because H and K are subgroups of G , and we deduce that x 1 H K . It now follows that H K is a subgroup of G as required. 2. 1.2.3 on page 27 Use the standard generators and relations to show that every ele ment of D 2 n which is not a power of r has order 2. Deduce that D 2 n is generated by the two elements s and sr , both of which have order 2. Each element of D 2 n can be written uniquely in the form r i or sr i , where 0 i n 1 (see page 25). The elements r i are of course powers of r , so we need to prove that sr i has order 2. Since sr i 6 = e , it will be sufficient to show that...
View Full
Document
 Spring '08
 Staff
 Math, Algebra

Click to edit the document details