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Let G = {±1, ±i, ±j, ±k}, where i? = j² = k² = -1, -i = (-1)i, 1² = (–1)² = 1, ij = -ji = k, jk = -kj = i, and ki = -ik = j. a. Show that H = {1, –1} < G.

Let G = {±1, ±i, ±j, ±k}, where i? = j² = k² = -1, -i = (-1)i, 1² = (–1)² = 1, ij = -ji = k, jk = -kj = i, and ki = -ik = j. a. Show that H = {1, –1} < G.

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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Let G = {±1, ±i, ±j, ±k}, where i? = j² = k² = –1, -i = (-1)i, 1² = (-1)² = 1, ij = -ji = k, jk = -kj = i, and ki = –ik = j. a. Show that H = {1, –1} < G. b. Construct the Cayley table for G/H. Is G/H isomorphic to Z, or Z,OZ? (The rules involving i, j, and k can be remembered by using the cir- cle below. k
Transcribed Image Text:Let G = {±1, ±i, ±j, ±k}, where i? = j² = k² = –1, -i = (-1)i, 1² = (-1)² = 1, ij = -ji = k, jk = -kj = i, and ki = –ik = j. a. Show that H = {1, –1} < G. b. Construct the Cayley table for G/H. Is G/H isomorphic to Z, or Z,OZ? (The rules involving i, j, and k can be remembered by using the cir- cle below. k
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