Discrete Mathematics - Homomorphism and Groups

[discretemaths]

I have two questions and I would be very appreciative if anyone could solve them for me. Here is the context for the questions:

Let (A,∗) be the semigroup A={0,1,2,3} and

0∗0 = 0
0∗1 = 0
0∗2 = 0
0∗3 = 0

1∗0 = 0
1∗1 = 1
1∗2 = 2
1∗3 = 3

2∗0 = 0
2∗1 = 2
2∗2 = 0
2∗3 = 2

3∗0 = 0
3∗1 = 3
3∗2 = 2
3∗3 = 1

Q1. In general, a homomorphism from a semigroup (A,∗) to a semigroup (B,~) is a function f:A→B such that f(a∗b) =f(a)~f(b).There are exactly six homomorphisms from (A,∗) to itself. Find all six of them.

Q2. Suppose that (B,~) is a semigroup. Let C be the set of invertible homomorphisms from (B,~) to itself. Let ◦ be composition of functions. Show that (C,◦) is a group.

 

[Steven G]

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