Solution Offered for Phoebe's Abstract Algebra Problem

Thanks to reader Caleb Shay, who "couldn't bear to see that poor girl suffer",  Phoebe's Abstract Algebra homework will be done on time!   Recall the statement of the problem:

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). Caleb's solution is as follows, originally appearing in an unpublished comment- but he preferred to have it displayed in the main blog (no problemo!):


"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."

Phoebe’s bonus problems:

1) Let the set S3 of cycle permutations be S3 = {I1(12), (23), (13), (123), (132)}

We establish an automorphism by mapping image to domain within the set S3 itself, such that – for example:

Φ(12) = (13)

Then show the other mappings within S3.

2) Proposition:  Φ _x  is a homomorphism.   Prove it!

Hint: Show that   Φ _x(y o z) = Φ _x  (y) o  Φ _x (z)