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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Abstract Algebra
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.
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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