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1 -1 16 ~ -j 1 -1 1 -1 1 -i i 1 j -j -k j k -k k -1 −j i i -2 -i i 1 LI -1 -i j -j -1 k k -k J.J. -j k - k k -k k j - k -k -j ງ k j - j 1 i j -j k -k 1 j -j -i -k -k k - j j i -i 1 1 −1 -1 (a) Show that Z = {±1} is the center of G. (b) Construct the table for G/Z. Is G/Z ≈ Z4 or Z₂ × Z₂ (and why)? (c) Show that G itself is not isomorphic to Z8, Z4 × Z2, Z2 × Z2 × Z2, or D4

1 -1 16 ~ -j 1 -1 1 -1 1 -i i 1 j -j -k j k -k k -1 −j i i -2 -i i 1 LI -1 -i j -j -1 k k -k J.J. -j k - k k -k k j - k -k -j ງ k j - j 1 i j -j k -k 1 j -j -i -k -k k - j j i -i 1 1 −1 -1 (a) Show that Z = {±1} is the center of G. (b) Construct the table for G/Z. Is G/Z ≈ Z4 or Z₂ × Z₂ (and why)? (c) Show that G itself is not isomorphic to Z8, Z4 × Z2, Z2 × Z2 × Z2, or D4

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.6: Variation
Problem 15E
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1 −1 i !!. 1 1 Ï ! i j j - j j k -k -1 -1 -j-j k j - j J - J i − j 1 k -k J.J. i -i i -i −1 1 -k k -i 1 1 1 j -j -i -k-k k -j j i k k j - k k -k k -j - k - k k j j - j -1 -i i −1 1 №. d. i 1 -1 (a) Show that Z = {±1} is the center of G. (b) Construct the table for G/Z. Is G/Z ≈ Z₁ or Z₂ × Z2 (and why)? (c) Show that G itself is not isomorphic to Z8, Z4 × Z2, Z2 × Z2 × Z2, or D4
Transcribed Image Text:1 −1 i !!. 1 1 Ï ! i j j - j j k -k -1 -1 -j-j k j - j J - J i − j 1 k -k J.J. i -i i -i −1 1 -k k -i 1 1 1 j -j -i -k-k k -j j i k k j - k k -k k -j - k - k k j j - j -1 -i i −1 1 №. d. i 1 -1 (a) Show that Z = {±1} is the center of G. (b) Construct the table for G/Z. Is G/Z ≈ Z₁ or Z₂ × Z2 (and why)? (c) Show that G itself is not isomorphic to Z8, Z4 × Z2, Z2 × Z2 × Z2, or D4
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