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. $$ \phi: D_{4} \rightarrow S_{5} $$

Short answer

A nontrivial homomorphism \( \phi: D_4 \rightarrow S_5 \) can map reflections to transpositions.

Step-by-step solution

Understand the Groups

The dihedral group \( D_4 \) is the symmetry group of a square, consisting of 8 elements: 4 rotations and 4 reflections. The symmetric group \( S_5 \) consists of all permutations of 5 elements with a total of 120 elements.

Define a Homomorphism

A homomorphism \( \phi: D_4 \rightarrow S_5 \) is a map such that for any two elements \( a, b \in D_4 \), \( \phi(ab) = \phi(a)\phi(b) \). We need a nontrivial homomorphism, meaning \( \phi \) is not the zero map (which sends all elements of \( D_4 \) to the identity element in \( S_5 \)).

Determine Kernel Possibilities

The order of the image of \( \phi \) is a divisor of \(|D_4| = 8 \). Moreover, the kernel of \( \phi \) should be a normal subgroup of \( D_4 \), and \(|\mathrm{ker}(\phi)| \times |\mathrm{im}(\phi)| = 8 \). Thus, \( |\mathrm{im}(\phi)|\) must be 1, 2, 4, or 8.

Choose a Simple Nontrivial Mapping

Let's construct a simple nontrivial map by assuming the kernel is the rotation subgroup of \( D_4 \). Map any non-identity element outside this kernel to an element of order 2 in \( S_5 \) such as a transposition. Map \( r \), a generator of the rotations, to \( \mathrm{id} \) and a reflection \( s_1 \) to \( (12) \). This mapping respects the group operation because \( s_1 \) and \( s_1 \cdot s_1 = \mathrm{id} \).

Verify Homomorphism Conditions

Check if the above mapping preserves operation: \( \phi(s_1^2) = \phi(e) = \mathrm{id} \) in \( S_5 \), and for any \( r^k \, s_1 \), \( \phi((r^k s_1)^2) = \mathrm{id} \). As this structure holds, the mapping is a valid homomorphism.

Key concepts

Dihedral Group

The Dihedral Group, denoted as \( D_4 \), is known for representing the symmetries of a square. It consists of eight elements: four rotations and four reflections. These transformations can fully express all the symmetries of a square.

• Rotations: Typically, they are denoted by \( r, r^2, r^3 \), where \( r \) is a rotation by 90 degrees, \( r^2 \) by 180 degrees, and \( r^3 \) by 270 degrees, with \( r^4 = e \), the identity element.
• Reflections: Notated as \( s_1, s_2, s_3, s_4 \), each representing a reflection across a line of symmetry through the center of the square.

Altogether, these elements form a group under the operation of composition. Each operation can be undone, adhering to the property of group invertibility. The group operation is associative, and there is an identity element and an inverse for every element.
Understanding \( D_4 \) is crucial for constructing homomorphisms involving geometric symmetries.

Symmetric Group

The Symmetric Group, denoted \( S_n \), is the group consisting of all possible permutations of \( n \) elements. For \( S_5 \), we are dealing with permutations of five elements which amounts to 120 unique permutations or arrangements of the elements.

• Elements: Each element in \( S_5 \) is a permutation, such as switching the first and second elements while leaving others unchanged, symbolized as \( (12) \).
• Order: The order, or total number of elements, in \( S_5 \) is 5 factorial (\( 5! = 120 \)). This results from every element being able to take any other position, generating extensive permutation possibilities.

Due to its ability to represent any transformation of 5 elements, \( S_5 \) is fundamental in understanding complex transformations and how group homomorphisms relate between different groups. It acts as a key structure in higher mathematical contexts such as Galois theory and algebraic topology.

Kernel of Homomorphism

The "Kernel of a Homomorphism" \( \phi: G \rightarrow H \) from one group\( G \) to another group \( H \) plays a central role in understanding group structures and their maps. Defined as the set of elements in \( G \) that map to the identity element in \( H \), it is a measure of how much of the group \( G \) gets "flattened" to the identity by the mapping \( \phi \).

• Denotation: The kernel is notated as \( \text{ker}(\phi) \).
• Properties: It is always a normal subgroup of \( G \), meaning it is invariant under conjugation by elements of \( G \).
• Relation to Image: The First Isomorphism Theorem tells us that the factor group \( G/\text{ker}(\phi) \) is isomorphic to the image of \( \phi \) in \( H \), establishing a crucial connection between the structure of the domain and range groups through the kernel.

In this exercise, the kernel of the homomorphism from \( D_4 \) to \( S_5 \) significantly helps to dictate what form the homomorphism can take.

Homomorphism Properties

When dealing with homomorphisms, several properties ensure that these mappings are well-defined and useful in exploring group relations and structures.

• Operation Preserving: For a homomorphism \( \phi: G \rightarrow H \), it must hold that for all elements \( a, b \in G \), the equation \( \phi(ab) = \phi(a)\phi(b) \) is satisfied. This property makes them essential in maintaining the structure of group operations even when transitioning from one group to another.
• Identity Map: A trivial homomorphism is one that sends every element of \( G \) to the identity element in \( H \), while nontrivial homomorphisms map elements onto more than just the identity.
• Kernel and Image: The size of the image is linked to the size of the kernel, and together with group orders, they satisfy the equation \( |\text{ker}(\phi)| \times |\text{im}(\phi)| = |G| \).

Homomorphisms are pivotal, as they help in classifying groups by their structure-preserving properties, and allow us to develop deeper insights into group theory through equivalences and mappings.