Question

In Exercises 23 through 30 construct an example of a nontrivial homomorphism between the two indicated groups, if this is possible, or explain why this is not possible. $$ \phi: S_{5} \rightarrow \mathbb{Z}_{2} $$

Short answer

No nontrivial homomorphism exists from \(S_5\) to \(\mathbb{Z}_2\).

Step-by-step solution

Understand the Groups Involved

The group \(S_5\) is the symmetric group on 5 elements, which consists of all permutations of the set \( \{1, 2, 3, 4, 5\} \), and has 120 elements. The group \(\mathbb{Z}_2\) is the cyclic group of order 2, with elements \(\{0, 1\}\) under addition modulo 2.

Determine Possible Homomorphism Existence

A homomorphism from a group \(G\) to a group \(H\) is a map \(\phi: G \rightarrow H\) that respects the group operation. For a homomorphism from \(S_5\) to \(\mathbb{Z}_2\), the image of \(\phi\) must be a subgroup of \(\mathbb{Z}_2\). \(\mathbb{Z}_2\)'s only subgroups are itself and the trivial group, since it is of prime order.

Consider the Kernel of the Homomorphism

To determine if a nontrivial homomorphism exists, note that the kernel of \(\phi\) must be a normal subgroup of \(S_5\). Also, the index of the kernel must divide the order of \(S_5\) (120) and also be a multiple of the order of the image (2) according to the First Isomorphism Theorem.

Identify Suitable Subgroups of \(S_5\)

The only normal subgroups of \(S_5\) that could work with a homomorphic map to \(\mathbb{Z}_2\) are \(\{ e \}\) (the trivial group) and \(A_5\) (the alternating group, which is not a candidate because its order is 60). Thus, the image of \(\phi\) being nontrivial implies a degree 2 subgroup, and \(S_5\) does not have such normal subgroups.

Conclude No Nontrivial Homomorphism Exists

Since there is no subgroup of \(S_5\) that can serve as the kernel for a nontrivial homomorphism to \(\mathbb{Z}_2\), no such homomorphism is possible. The only homomorphism from \(S_5\) to \(\mathbb{Z}_2\) is the trivial one where every element maps to 0.

Key concepts

Symmetric Group

Symmetric groups, denoted as \(S_n\), are an essential concept in group theory. These are groups made up of all possible permutations of \(n\) elements. For instance, \(S_5\) is the symmetric group on the set \(\{1, 2, 3, 4, 5\}\), consisting of every possible ordering or permutation of these five numbers. This set includes 120 distinct permutations because the factorial of five, \(5!\), equals 120.

Each element in \(S_n\) uniquely shifts the elements of the set, implementing the function "rearrange in new order." Therefore, symmetric groups are pivotal in studying how objects can be arranged or ordered. The operation in these groups is function composition, meaning we apply one permutation then another, fast and clean like stacking tasks in a row.

Understanding \(S_5\) and its properties not only expands knowledge about symmetries and permutations but also aids in examining the structures of other mathematical entities.

Cyclic Group

A cyclic group is a special type of group that can be generated entirely by repeatedly applying an operation to a single element within the group. For each element \(g\) in a cyclic group, \(\{ e, g, g^2, g^3, \ldots \}\) captures all the group's elements through using the operation, typically addition or multiplication.

For example, \(\mathbb{Z}_2\) is a simple cyclic group formed by \(\{0, 1\}\) and addition modulo 2. Here, \(0\) serves as the identity element, and \(1 + 1 = 0\), showing the loop back to the starting point, making this group "cyclic."

This characteristic simplicity makes cyclic groups a fundamental and building block in the larger world of group theory, offering clear, controlled examples to explore more complex mathematical ideas.

Normal Subgroup

In group theory, a subgroup \(N\) of a group \(G\) is said to be "normal" if it remains invariant under conjugation by elements of \(G\). In simpler terms, if you pick any element in \(G\), and any element in \(N\), the equation \(gng^{-1} \in N\) holds; meaning rearranging elements in that special way keeps the set within \(N\). The notation for a normal subgroup is \(N \triangleleft G\).

Normal subgroups are crucial because they allow us to partition \(G\) into factor groups, and are essential when exploring homomorphisms and complex group structures. Without normal subgroups, constructing meaningful and robust homomorphisms becomes an arduous task because they serve as the kernel in these maps, allowing us to explore the structure of the larger group through simpler pieces.

Kernel of a Homomorphism

The kernel of a group homomorphism \(\phi: G \rightarrow H\) is the set of elements in \(G\) that map to the identity element in \(H\). Essentially, it captures which elements become "invisible" under the homomorphic map, acting as a measure of what 'falls to zero' in the target group.

The kernel \(\text{ker}(\phi)\) is a normal subgroup of \(G\). This is important because the First Isomorphism Theorem states that \(G/\text{ker}(\phi)\) is isomorphic to the image of \(\phi\). Thus, understanding the kernel gives insights into both the structure and function of the homomorphism, offering a bridge between the domain and codomain by showcasing transformations from one to another.

The kernel plays a vital role in determining whether the homomorphism can be trivial or not, such as the impossible task of mapping \(S_5\) nontrivially into \(\mathbb{Z}_2\) in the given exercise.