Write the Cayley tables for addition and multiplication in #ZZ_7#?

1 Answer
George C.
Feb 24, 2018

Here are the Cayley tables for addition and multiplication in #ZZ_7#...

Explanation:

Cayley tables are two dimensional grids describing the results of addition or multiplication of all elements in a group. In the case of a ring such as #ZZ_7#, there are separate tables for addition and multiplication.

Here is the table for addition:

#color(white)(" " 0 " ")underline(color(white)("| ") 0 color(white)(" | ") 1 color(white)(" | ") 2 color(white)(" | ") 3 color(white)(" | ") 4 color(white)(" | ") 5 color(white)(" | ") 6 color(white)(" |"))#
#" " 0 color(white)(" ")underline("| " 0 " | " 1 " | " 2 " | " 3 " | " 4 " | " 5 " | " 6 " |")#
#" " 1 color(white)(" ")underline("| " 1 " | " 2 " | " 3 " | " 4 " | " 5 " | " 6 " | " 0 " |")#
#" " 2 color(white)(" ")underline("| " 2 " | " 3 " | " 4 " | " 5 " | " 6 " | " 0 " | " 1 " |")#
#" " 3 color(white)(" ")underline("| " 3 " | " 4 " | " 5 " | " 6 " | " 0 " | " 1 " | " 2 " |")#
#" " 4 color(white)(" ")underline("| " 4 " | " 5 " | " 6 " | " 0 " | " 1 " | " 2 " | " 3 " |")#
#" " 5 color(white)(" ")underline("| " 5 " | " 6 " | " 0 " | " 1 " | " 2 " | " 3 " | " 4 " |")#
#" " 6 color(white)(" ")underline("| " 6 " | " 0 " | " 1 " | " 2 " | " 3 " | " 4 " | " 5 " |")#

Here is the table for multiplication of non-zero elements:

#color(white)(" " 0 " ")underline(color(white)("| ") 1 color(white)(" | ") 2 color(white)(" | ") 3 color(white)(" | ") 4 color(white)(" | ") 5 color(white)(" | ") 6 color(white)(" |"))#
#" " 1 color(white)(" ")underline("| " 1 " | " 2 " | " 3 " | " 4 " | " 5 " | " 6 " |")#
#" " 2 color(white)(" ")underline("| " 2 " | " 4 " | " 6 " | " 1 " | " 3 " | " 5 " |")#
#" " 3 color(white)(" ")underline("| " 3 " | " 6 " | " 2 " | " 5 " | " 1 " | " 4 " |")#
#" " 4 color(white)(" ")underline("| " 4 " | " 1 " | " 5 " | " 2 " | " 6 " | " 3 " |")#
#" " 5 color(white)(" ")underline("| " 5 " | " 3 " | " 1 " | " 6 " | " 4 " | " 2 " |")#
#" " 6 color(white)(" ")underline("| " 6 " | " 5 " | " 4 " | " 3 " | " 2 " | " 1 " |")#

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