Abstract Algebra Problem Solution

  Pr0blem:  Prepare the respective tables for the isomorphism and give specific examples in terms of the function φ, i.e. show specific mappings. (Where: φ(x) φ(y) = φ(xy) for example).

Solution:

The solution included preparation of the respective addition table (in  Z₄ ) and multiplication table (in  Z₁₀ ). The addition table for Z4 is (see e.g. Fig. 1 in http://www.brane-space.blogspot.com/2012/11/looking-again-at-groups-sub-groups.html ):

+ /----0 ----1 ---2 ----3
-----------------------
0 --- 0------1 ---2 ----3

1 -----1 -----2 ---3---- 0

2 -----2 -----3--- 0 ----1

3----- 3 -----0 ---1---- 2
-----------------------------

The multiplication table in Z₁₀


-*--/----6 ----2 -----4 ----8
---------------------------
6 --- ---6------2 --- 4 ----8

2 ------2 ------4 ----8----6

4 ------4 ------8--- -6 ---2

8----- 8 -----6 ----2---- -4
-----------------------------

Since 6 *6 = 6 then the identity element in  Z₁₀  is 6.. 

So φ(x) φ(x) = φ(x^2) = (φ6)φ(6) =φ( I)

Then the mapping isomorphisms between the tables will be:

Φ: 0 -> 6 (identity to identity element)

Φ: 2 -> 4

Φ: 3 -> 8

Φ: 1 -> 2


So each of the numbers on the left side of the arrow is mapped to the corresponding number on the right side, and this is done using the two tables such that each element of Z₄  is mapped to each element of Z₁₀.




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