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There are two group of order 4, namely Z, and Z2 Z2, and only one of them can be isomorphic to G/Z. Fill in the the multiplication table for the quotient group G/Z, and determine which of the two groups of order 4 is isomorphic to G/Z * || Z iz z iz jz kZ iZ jZ kZ

There are two group of order 4, namely Z, and Z2 Z2, and only one of them can be isomorphic to G/Z. Fill in the the multiplication table for the quotient group G/Z, and determine which of the two groups of order 4 is isomorphic to G/Z * || Z iz z iz jz kZ iZ jZ kZ

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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Below is the multiplication table for G=the "octonion group" 1 -1 -i 1 j -j k -k * 1 1 -1 -i k -k -j -j -1 -1 1 -i i ーk k i -1 1 k -k -j -i -i i 1 -1 -k k j -j ーj k ーk k -1 1 i -j -j -k 1 -1 -i i k k ーk ーj -i i -1 1 -k ーk k -j i -i 1 -1 Let Z = {1, –1} be the center. There are four left cosets of Z, namely • 1Z = Z = {1, –1}, whose elements are shown in red iZ = {i, -i}, whose elements are shown in gray • jZ = {j,-j}, whose elements are shown in olive • kZ = {k, –k}, whose elements are shown in blue.
Transcribed Image Text:Below is the multiplication table for G=the "octonion group" 1 -1 -i 1 j -j k -k * 1 1 -1 -i k -k -j -j -1 -1 1 -i i ーk k i -1 1 k -k -j -i -i i 1 -1 -k k j -j ーj k ーk k -1 1 i -j -j -k 1 -1 -i i k k ーk ーj -i i -1 1 -k ーk k -j i -i 1 -1 Let Z = {1, –1} be the center. There are four left cosets of Z, namely • 1Z = Z = {1, –1}, whose elements are shown in red iZ = {i, -i}, whose elements are shown in gray • jZ = {j,-j}, whose elements are shown in olive • kZ = {k, –k}, whose elements are shown in blue.
There are two group of order 4, namely Z4 and Z2 Z2, and only one of them can be isomorphic to G/Z. Fill in the the multiplication table for the quotient group G/Z, and determine which of the two groups of order 4 is isomorphic to G/Z * || z iz jZ kZ Z iz jZ kZ
Transcribed Image Text:There are two group of order 4, namely Z4 and Z2 Z2, and only one of them can be isomorphic to G/Z. Fill in the the multiplication table for the quotient group G/Z, and determine which of the two groups of order 4 is isomorphic to G/Z * || z iz jZ kZ Z iz jZ kZ
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