Combinatorial Group Theory: Presentations of Groups in Terms by Wilhelm Magnus PDF

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Example text

4 1. Show that if F is the free group on xl' x z, ... , Xn and U is cyclically reduced, then Uk is also cyclically reduced. Show that if '1' = RU R-l is freely reduced, and U is cyclically reduced, then p(Tk) = RUkR-l. 2. Show that if U and V are freely reduced and Uk "" Vk, k an integer oft 0, then U = V. [Hint: If U is cyclically reduced but V is not, then, by Problem 1, p( Uk) = Uk is cyclically reduced but p( Vk) is not. Hence, if U is cyclically reduced, then V is cyclically reduced and Uk = Vk; thus U = V.

1. 3 1. Solve the word and transformation problem for (a, b; as, b2 , ab = ba- I ). ] 2. Solve the word and transformation problem for (a, b; as, b4 , ab = ba- I ). ] 3. Show that a finite multiplication table presentation has a solvable word problem and transformation problem. 1 for the word problem. ] 4. Let G = (aI' a 2 • •••• an; R I , R 2 , •••• R m ). and let < be the relation of precedes defined in this section. :S; n; then show that the number of words of length r which precede W is a;;, where P(fi) is the number of generating symbols or their inverses preceding Ii' 5.

B- Po , or b-Poa-ar+l, where f3o, oc l ' f31' ••• , OCT' f3ro OCT+I are non·zero integers. lbPt ... a·rbiJr , and can define the identity element. (h) Conclude that one set of canonical forms for (a, b) consists of the empty word, together with the words of the form where oc l or {Jr may be zero, but otherwise all OCi and fJi are non-zero integers. This solves the word problem for the free group (a, b). (i) Conclude also that (a, b) is a presentation for the group generated by and 32 SEC. 3 10.

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