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ahw2 - Math 4124 Wednesday February 1 Second Homework...

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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 = e , it will be sufficient to show that ( sr i ) 2 = e ; we shall show
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