Here is problem 17 from 1.2. Let X_2n be the group whose presentation is displayed in (1.2), i.e., <x,y|xⁿ = y² = 1, x y = y x²>.

(a) Show that if n = 3k, then X_2n has order 6, and it has the same generators and relations as D₆ when x is replaced by r and y by s.

(b) Show that if (3,n)=1, then x satisfies the additional relation: x=1. In this case deduce that X_2n has order 2. [Use the facts that xⁿ = 1 and x³ = 1.]

Here is problem 18 from 1.2. Let Y be the group whose presentation is displayed in (1.3), i.e., Y = <u,v|u⁴ = v³ = 1, u v = v² u²>.

(a) Show that v² = v^-1. [Use the relation v³ = 1.]

(b) Show that v commutes with u³. [Show that v² u³ v = u³ by writing the left hand side as (v² u²)(u v) and using the relations to reduce this to the right hand side. Then use part (a).]

(c) Show that v commutes with u. Show that u⁹ = u and then use part (b).]

(d) Show that u v = 1. [Use part (c) and the last relation.]

(e) Show that u = 1, deduce that v = 1, and conclude that Y = 1. [Use part (d) and the equation u⁴ v³ = 1.]