Homomorphism on modulo groups

[raynard]

I was wondering, if we want to define a morphism from
$$\mathbb{Z}$$₂₀₀₆ to, let's say $$\mathbb{Z}$$₃₀₀₈.
Obviously, all linear functions like $$x \rightarrow a\cdot x$$ will do, but are there any other functions which can result in a homomorphism?

 

[Outlined]

Well, in the case of this cyclic group an homomorphism depends on the generating element of the group. When you have the image of your generating element you have the image of every other element of the group. The only restriction is

image_generating_element²⁰⁰⁶ = 0.

In your case the claim that all homom. will do is FALSE. Example, define

f : Z₂₀₀₆ -> Z₃₀₀₈
x |-> 20x.

Apply f to 2006 = 1 + 1 + ... + 1:

1016 = 40120 = 20 + 20 + ... + 20 _{[2006 times]}= f(1) + f(1) + ... + f(1) _{[2006 times]}= f(1 + 1 + ... + 1 _{[2006 times]}) = f(2006) = f(0) = 0

CONTRADICTION

 

[raynard]

I understand my error in believing every linair function will result in a homomorphism.
However, I don't fully understand how the restriction on a generator (gen²⁰⁰⁶ = 0) can help in finding function that do result in a morphism. The only possible combination is the trivial solution x |-> x.

 

[Outlined]

Well, sometimes a homomorphism doesn't even exists (except for the trivial homom. which maps everything to 0), for example, try to find a homom. between Z₆ and Z₇ (in which EVERY element is of order 7, as 7 is prime).

 

[blue_raver22]

A homomorphism is a function that preserves the algebraic structure of a group. In the case of modulo groups, the algebraic structure is the operation of addition and the preservation of this structure means that the function must preserve the addition operation.

In the case of \mathbb{Z}2006 and \mathbb{Z}3008, both groups have the same operation of addition (mod 2006 in the first group and mod 3008 in the second group). Therefore, any function that preserves the addition operation will result in a homomorphism.

Aside from linear functions, there are other functions that can result in a homomorphism between these two groups. For example, the function $f(x) = x^2$ will also preserve the addition operation and therefore, will result in a homomorphism.

In general, any function that satisfies the property $f(x+y) = f(x) + f(y)$ will result in a homomorphism between modulo groups. This is because this property ensures that the function preserves the addition operation.

In conclusion, while linear functions are the most obvious and common examples of homomorphisms between modulo groups, there are other functions that can also result in a homomorphism as long as they satisfy the property of preserving the addition operation.