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Let n be an even positive integer. In this exercise, we will construct a new family of finite groups Qn of order 2n using the quaternions. (a) Show that the quaternions w = cos + i sin 2 and j satisfy the following n three relations: 2π n w" = 1₁ j² = w³² , jw = w ¹j. (b) Let Qn be the group generated by {w, j} under the quaternion multiplication. Show that Qnl 2n. More precisely, show that = Qn= {1, w, w²₁. , wn-¹, j, wj, w ² j, ..., wn-1, ¹j} = {1,w₁w²,. , wn-1, j, jw, jw², ..., jwn-¹}. This is called the dicyclic group of order 2n. Show that the dicyclic group of order 8 is exactly the quaternion group {±1, ti, tj, ±k}. (c) Show that QnDn.

Let n be an even positive integer. In this exercise, we will construct a new family of finite groups Qn of order 2n using the quaternions. (a) Show that the quaternions w = cos + i sin 2 and j satisfy the following n three relations: 2π n w" = 1₁ j² = w³² , jw = w ¹j. (b) Let Qn be the group generated by {w, j} under the quaternion multiplication. Show that Qnl 2n. More precisely, show that = Qn= {1, w, w²₁. , wn-¹, j, wj, w ² j, ..., wn-1, ¹j} = {1,w₁w²,. , wn-1, j, jw, jw², ..., jwn-¹}. This is called the dicyclic group of order 2n. Show that the dicyclic group of order 8 is exactly the quaternion group {±1, ti, tj, ±k}. (c) Show that QnDn.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. Let n be an even positive integer. In this exercise, we will construct a new family of finite groups Qn of order 2n using the quaternions. 2π (a) Show that the quaternions w = cos ² + i sin 2 and j satisfy the following three relations: n n n w² = 1₁ j² = w²2², jw =w¯¹j. 1, (b) Let Qn be the group generated by {w, j} under the quaternion multiplication. Show that |n| = 2n. More precisely, show that Qn = {1,w₁w², wn-1, j, w j, w ²j, = {1,w, w²,...,w²-1, j, jw, jw², ... 9 an ... 9 This is called the dicyclic group of order 2n. Show that the dicyclic group of order 8 is exactly the quaternion group {±1, ti, tj, ±k}. n = 1, 6² a, ba - = (c) Show that Qn Z Dn. (d) Let G be a finite group of order 2n. Show that if G can be generated by two elements {a,b} which satisfy the relations 'j} jwn-1}. = a ¹b, then GQn. (Hint: First show that G = {1, a, ..., an-1, b, ab, ..., an-¹b}. Then show that the Cayley table of G is completely determined by the three relations of a and b. Conclude that G has exactly the same Cayley table as that of Qn, and hence GQn.)
Transcribed Image Text:3. Let n be an even positive integer. In this exercise, we will construct a new family of finite groups Qn of order 2n using the quaternions. 2π (a) Show that the quaternions w = cos ² + i sin 2 and j satisfy the following three relations: n n n w² = 1₁ j² = w²2², jw =w¯¹j. 1, (b) Let Qn be the group generated by {w, j} under the quaternion multiplication. Show that |n| = 2n. More precisely, show that Qn = {1,w₁w², wn-1, j, w j, w ²j, = {1,w, w²,...,w²-1, j, jw, jw², ... 9 an ... 9 This is called the dicyclic group of order 2n. Show that the dicyclic group of order 8 is exactly the quaternion group {±1, ti, tj, ±k}. n = 1, 6² a, ba - = (c) Show that Qn Z Dn. (d) Let G be a finite group of order 2n. Show that if G can be generated by two elements {a,b} which satisfy the relations 'j} jwn-1}. = a ¹b, then GQn. (Hint: First show that G = {1, a, ..., an-1, b, ab, ..., an-¹b}. Then show that the Cayley table of G is completely determined by the three relations of a and b. Conclude that G has exactly the same Cayley table as that of Qn, and hence GQn.)
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