SOLUTION: This a Abstract Algebra class Let G=⟨a⟩ be a cyclic group of order10. Define φ:Z→G by φ(n)=a^n for all n∈Z. Assume that φ is a homomorphism. Find kerφ.

Question 1186053: This a Abstract Algebra class
Let G=⟨a⟩ be a cyclic group of order10. Define φ:Z→G by φ(n)=a^n for all n∈Z. Assume that φ is a homomorphism. Find kerφ.
Answer by ikleyn(52932) (Show Source): You can put this solution on YOUR website!
.
This a Abstract Algebra class
Let G=⟨a⟩ be a cyclic group of order10. Define φ:Z→G by φ(n)=a^n for all n∈Z.
Assume that φ is a homomorphism. Find kerφ.
~~~~~~~~~~~~~~~`

The group G is presented as  G = (a) in the post.

Also, from the context, G is a multiplicative group, regarding its operation.



It means that element "a" of the group G is the generator of the cyclic group G.



In turn, it means that   = 1  (the unit element of the group G),  

while  the elements    at k = 1, 2, 3, . . . , 9 are all different and not equal to the unit element of the group.



Hence, kerφ is the set of all integers in Z that are multiples of 10:   { . . . , -20, -10, 0, 10, 20, . . . }.    ANSWER

Solved and explained.

===============

We should not "assume" that φ , defined in this way, is a homomorphism.

It REALLY IS the homomorphism.

RELATED QUESTIONS
From Abstract Algebra class. Consider Z9 - { [ 0 ] } with respect to multiplication [... (answered by ikleyn)

Let n be a fixed positive integer in Z. Define the relation ≡n on Z by x ≡n y if and (answered by math_tutor2020)

This for a Abstract Algebra class Assume that φ is a homomorphism from the group G to (answered by robertb)

I have a test this Thursday and I dont understand this review question at all. Can... (answered by jsmallt9)

Your help will be appreciated thanks! Let a(1)=5. Define a(n) to be (a(n-1))/2 if a(n-1) (answered by ikleyn,Theo)

Let S = {a, e, j, n, t, x} be a sample space of an experiment and let F = {a, n, x} and G (answered by Fombitz)

if f is charater of a group G of order n then how for a belongs to G... (answered by richwmiller)

using this data 3,13,15,16,16,20,23 N=7 what score would a z of -.25 be and what score... (answered by stanbon)

Let A = {n C Z | n is odd} and B = {n C Z | n^2 - 1 mod 4}. Prove that A is included in... (answered by ikleyn)