Question

Find all possible nontrivial homomorphisms between the indicated groups. $$ \phi: D_{4} \rightarrow \mathbb{Z}_{4} $$

Short answer

Nontrivial homomorphisms are \( \phi(r) = 0,1,2,3 \) with \( \phi(s) = 2 \).

Step-by-step solution

Understand Group Structures

The group \( D_4 \) is the dihedral group of order 8, representing symmetries of a square, which contains rotations and reflections. It can be generated by the elements \( r \) and \( s \) where \( r^4 = e \), \( s^2 = e \), and \( srs = r^{-1} \). Meanwhile, \( \mathbb{Z}_{4} \) is the cyclic group of integers modulo 4 and is generated by the element 1.

Identify Homomorphism Conditions

A homomorphism \( \phi: D_4 \to \mathbb{Z}_4 \) must satisfy two key conditions: \( \phi(r^4) = 0 \) (identity in \( \mathbb{Z}_4 \)) and \( \phi(s^2) = 0 \). The homomorphism must respect group operation structure: \( \phi(rs) = \phi(r) + \phi(s) \).

Analyze Possible Assignments for Generators

Since \( \mathbb{Z}_4 \) has four elements, \( \phi(r) \) could be 0, 1, 2, or 3. For each choice, \( \phi(r^4) = 4\phi(r) = 0 \mod 4\), this implies all values for \( \phi(r) \) are valid. Now check \( \phi(s) \), which must satisfy \( 2\phi(s) = 0 \) in \( \mathbb{Z}_4 \); hence, \( \phi(s) \) can be 0 or 2.

Determine Nontrivial Cases

A trivial homomorphism sends every group element to the identity (0 in \( \mathbb{Z}_4 \)). To find nontrivial homomorphisms, avoid mapping both generators to 0 unless it's the zero homomorphism. Given that \( \phi(s) = 2 \) results in nonzero \( \phi \), check combinations with \( \phi(r) = 0, 1, 2, 3 \) to ensure valid mappings.

Verify Homomorphism Properties

For each valid pair \((\phi(r), \phi(s))\), verify the homomorphism properties. For example, if \( \phi(r) = 0 \) and \( \phi(s) = 2 \), then \( \phi(rs) = \phi(r) + \phi(s) = 2 \), which is consistent under \( D_4 \) and \( \mathbb{Z}_4 \) structure.

List All Nontrivial Homomorphisms

For valid \( \phi(s) = 2 \), the possible nontrivial homomorphisms are where \( \phi(r) \) can take any value from 0 to 3, maintaining \( \phi(s) = 2 \). These include mappings like: \( (0, 2) \), \( (1, 2) \), \( (2, 2) \), and \( (3, 2) \). Each represents a distinct nontrivial homomorphism as they map elements of \( D_4 \) into \( \mathbb{Z}_4 \) respecting group operations.

Key concepts

Dihedral Group: Understanding the Basics

The dihedral group, denoted as \( D_n \), is a fundamental concept in group theory. It represents the symmetries of a regular polygon with \( n \) sides. For the specific case of \( D_4 \), the group deals with the symmetries of a square. This includes both rotations and reflections.

• Rotations: There are four possible rotations: 0 degrees (identity), 90 degrees, 180 degrees, and 270 degrees.
• Reflections: There are four reflectional symmetries: two along the diagonals and two along the midlines of the square.

All these symmetries, when combined, form the dihedral group \( D_4 \) which has an order of 8 elements. The group can be generated using two elements: \( r \) for rotation and \( s \) for reflection. The properties of these generators are vital: \( r^4 = e \) and \( s^2 = e \), where \( e \) is the identity element. These fundamentals ensure that any combination of these operations brings the square back to a symmetrical position.

Cyclic Group: Building Simple Structures

A cyclic group is another foundational structure in group theory, and it is one where all elements are powers of a particular element, known as the generator. The group \( \mathbb{Z}_n \) is a common example, representing integers modulo \( n \).

For \( \mathbb{Z}_4 \), this is the group of integers under addition modulo 4, generated by the number 1. The elements of \( \mathbb{Z}_4 \) are \( 0, 1, 2, \) and \( 3 \). Each element can be visualized as a position on a clock face, where moving around corresponds to adding integers mod 4.

The cyclic nature of \( \mathbb{Z}_4 \) ensures that for any element \( x \), there exists another element such that the sum goes back to the identity, zero. This simple yet powerful structure allows for seamless mapping of symmetries seen in dihedral groups through homomorphisms.

Symmetries of a Square: A Closer Look

Symmetries of a square are naturally captured by the dihedral group \( D_4 \). Understanding these symmetries is crucial when studying group homomorphisms as they provide a framework to translate symmetrical actions into algebraic expressions.

• Rotational Symmetries: Each rotation aligns the square back to its initial form after completing full cycles, with the generator \( r \) creating clockwise moves.
• Reflectional Symmetries: Such as those flipping the square over an axis, follow the generator \( s \), displaying the properties of reflections.

These symmetries are more than just geometric transformations; they form a mathematical structure that is exhaustively represented by \( D_4 \). By understanding these symmetries, students can see how algebraic equations reflect geometric realities, simplifying complex tasks like defining nontrivial homomorphisms.

Nontrivial Homomorphisms: Moving Beyond Triviality

In group theory, a homomorphism is a function between two groups that respects the group operations. When considering nontrivial homomorphisms, it refers to those that map at least one element of the group to a non-identity element.

A homomorphism from \( D_4 \) to \( \mathbb{Z}_4 \), for example, should not merely map every element to zero in \( \mathbb{Z}_4 \) (which would be trivial). Instead, it needs to map some elements to non-zero values while following the group's operation conditions.

Finding a nontrivial homomorphism involves mapping each generator of \( D_4 \) to non-identity elements in \( \mathbb{Z}_4 \). Specifically, suppose we choose \( \phi(s) = 2 \) in a nontrivial setting. Such mappings ensure that the operation between elements in \( D_4 \) (e.g., rotations and reflections) correlates to additions in \( \mathbb{Z}_4 \), hence keeping the homomorphism intact.