Homomorphism from Z into a nonabelian group

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In summary, two potential examples of nontrivial homomorphisms from Z into a nonabelian group are f(n) = g^n and f(n) = g^n * h, where g and h are elements in a nonabelian group such as the diedral group D_3 of order 6. These homomorphisms demonstrate that there exist elements in the nonabelian group that do not commute, as desired. Additionally, it is noted that a homomorphism from Z into any group can be determined by choosing f(1), allowing for flexibility in selecting elements in the target group.
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quasar987
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What would be an example of a (nontrivial) homomorphism from Z into a nonabelian group??

More specifically, I am looking for two homomorphisms f, g: Z-->W such that for some n,m in Z f(n)g(m) does not commute.
 
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Ah! I found out that the smallest nonabelian group is the diedral group of order 6, D_3. And from its multiplication table, we see that embedded in it is a copy of Z_2 and a copy of Z_3 that contain elements that do not commute.
 
  • #3
A homomorphism from Z into any group G is completely specified by picking f(1), because if f(1)=g, then f(n)=gn. And you can pick f(1) to be any element of G that you want, basically because Z is a free group.
 
  • #4
Thanks for that remark StatusX.
 

Related to Homomorphism from Z into a nonabelian group

What is a homomorphism from Z into a nonabelian group?

A homomorphism from Z into a nonabelian group is a function that preserves the group structure between the integers and the nonabelian group. This means that the operation of the nonabelian group is preserved when the function is applied to the integers.

How is a homomorphism from Z into a nonabelian group different from a homomorphism into an abelian group?

A homomorphism from Z into a nonabelian group is different from a homomorphism into an abelian group because in a nonabelian group, the order of the elements matters when performing the group operation. This means that the function must preserve the order of the integers in addition to the group operation.

Can a homomorphism from Z into a nonabelian group be surjective?

Yes, a homomorphism from Z into a nonabelian group can be surjective, meaning that every element in the nonabelian group has a corresponding integer that maps to it. However, it is not always the case and may depend on the specific nonabelian group.

What is an example of a nonabelian group that can have a homomorphism from Z into it?

One example of a nonabelian group that can have a homomorphism from Z into it is the dihedral group, which is the group of symmetries of a regular polygon. The dihedral group is nonabelian because the order of the symmetries matters when combining them, and it can have a homomorphism from Z into it that maps integers to rotations of the polygon.

How are homomorphisms from Z into a nonabelian group useful in mathematics?

Homomorphisms from Z into a nonabelian group are useful in mathematics because they allow us to study the structure and properties of nonabelian groups by relating them to the familiar integers. They also have applications in other areas such as cryptography and physics.

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