Question

Find all possible nontrivial homomorphisms between the indicated groups. $$ \phi: \mathbb{Z}_{5} \rightarrow \mathbb{Z}_{10} $$

Short answer

No nontrivial homomorphisms exist between \(\mathbb{Z}_5\) and \(\mathbb{Z}_{10}\).

Step-by-step solution

Understand Homomorphism Definition

A homomorphism between two groups \(G\) and \(H\) is a function \(\phi: G \rightarrow H\) such that for all elements \(a, b \in G\), \(\phi(a + b) = \phi(a) + \phi(b)\). This means the group operation is preserved.

Identify Group Elements

Identify the elements of the groups: \(\mathbb{Z}_5 = \{0, 1, 2, 3, 4\}\) and \(\mathbb{Z}_{10} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). These groups are under modulo addition.

Determine Kernel of Homomorphism

Since homomorphisms map the identity element of the domain to the identity element of the codomain, \(\phi(0) = 0\). The kernel is the set of elements in \(\mathbb{Z}_5\) that map to 0 in \(\mathbb{Z}_{10}\). In this case, the kernel can be any subgroup of \(\mathbb{Z}_5\). Nontrivial homomorphisms require nontrivial kernels.

Evaluate Kernel Possibilities

Since the order of \(\mathbb{Z}_{10}\) is 10, a receiving element from \(\mathbb{Z}_5\) must divide 10 due to Lagrange's theorem. In \(\mathbb{Z}_5\), non-zero elements such as \(1, 2, 3, 4\) should map to some element in \(\mathbb{Z}_{10}\) in a way that is consistent with group operation rules and kernel consideration.

Consider Group Order Constraints

For homomorphisms to exist, the order of \(\phi(1)\) in \(\mathbb{Z}_{10}\) must divide the order of 1 in \(\mathbb{Z}_5\), which is 5. Thus, \(\phi(1)\) must be of order dividing 5 in \(\mathbb{Z}_{10}\). However, only \(0\) and \(5\) have orders that divide 5.

Establish Nontrivial Solutions

Since \(\mathbb{Z}_{10}\) does not have elements of order 5 and the only allowable kernel is trivial, there are no nontrivial homomorphisms that satisfy these requirements, as \(\phi(1) = 5\) cannot produce bijective or valid nontrivial homomorphisms because \(5x\) in \(\mathbb{Z}_{10}\) results in trivial images.

Key concepts

Elements of Cyclic Groups

When discussing cyclic groups, we first recognize that these groups are generated by a single element. This means that every element in the group can be expressed as some power or product of this generator. For instance, the group \( \mathbb{Z}_5 \) is a cyclic group generated by the element \( 1 \).
Using the generator, \( \mathbb{Z}_5 \) comprises elements \( \{0, 1, 2, 3, 4\} \). Each element here is essentially a repeated addition of the generator within the constraints of modulo 5 addition.
In the same breath, \( \mathbb{Z}_{10} \) is also cyclic, generated by \( 1 \), and contains elements \( \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \). Understanding the structure and generation of elements in cyclic groups is crucial because it helps determine possible homomorphic mappings between groups.

Kernel of Homomorphism

In group theory, the kernel of a homomorphism is a set of elements from the domain mapped to the identity in the codomain. For a homomorphism \( \phi: G \rightarrow H \), the kernel \( \ker(\phi) \) is the subset \( \{g \in G \mid \phi(g) = e_H \} \), where \( e_H \) is the identity of the group \( H \).
The kernel is significant since it can provide insights into the structure of the homomorphism. In our example, the requirement is for nontrivial homomorphisms, which implies we're interested in kernels that do not equal the entire group, but rather a true subgroup of \( \mathbb{Z}_5 \) like \( \{0\} \) (trivial) or any larger subset that fulfills the kernel definition.
However, in the homomorphism from \( \mathbb{Z}_5 \) to \( \mathbb{Z}_{10} \), nontrivial kernels would imply that some non-zero elements would map to zero, violating the constraint due to lack of suitable corresponding elements in \( \mathbb{Z}_{10} \). This attribute of kernels is important for determining the very possibility of certain homomorphisms.

Nontrivial Homomorphisms

Nontrivial homomorphisms are those where the mapping truly relates the structures of both groups, going beyond mapping everything trivially to the zero element. This is contrasted with a trivial homomorphism where every element maps to zero in the codomain.
For a nontrivial homomorphism between \( \mathbb{Z}_5 \) and \( \mathbb{Z}_{10} \), we would be looking for a scenario where the function \( \phi: \mathbb{Z}_5 \rightarrow \mathbb{Z}_{10} \) does not map all elements of \( \mathbb{Z}_5 \) to zero unless naturally dictated by the cyclic nature and modulo constraints.
However, since \( \mathbb{Z}_{10} \) lacks elements of order 5 relevant to \( \mathbb{Z}_5 \), establishing a nontrivial relationship becomes difficult. While \( 5 \) in \( \mathbb{Z}_{10} \) fits the order requirement, it results in trivial outputs rather than a meaningful homomorphic structure.

Lagrange's Theorem

Lagrange's theorem helps simplify and direct thoughts in group homomorphisms. It states that the order (number of elements) of any subgroup \( H \) of a finite group \( G \) divides the order of \( G \). This is crucial when assessing possible outputs in a homomorphic image.
In the case at hand, evaluating \( \phi : \mathbb{Z}_5 \rightarrow \mathbb{Z}_{10} \), lacks structuring where non-zero elements maintain a nom-residual value output.
Therefore, according to Lagrange’s theorem, the outputs of the homomorphism must have orders (divisors) of the group \( \mathbb{Z}_{10} \), such as divisors of 10. That means distinct orders such as \( 1 \) and \( 5 \) must be observed, but they map trivial kernels when tested against each structurally.