Define a homomorphism [tex]D_8 \rightarrow S_4[/tex] by labeling vertices of a square, as we did for a triangle. List the 8 permutations in the image of this homomorphism.
Answer (2)
Label the vertices of the square as 1, 2, 3, 4.
Identify the elements of D 8 : identity, rotations by 90, 180, and 270 degrees, and reflections about the horizontal, vertical, and two diagonal axes.
Determine the corresponding permutation of the vertices for each element of D 8 .
The 8 permutations in the image are: (), (1 2 3 4), (1 3)(2 4), (1 4 3 2), (2 4), (1 3), (2 3), (1 4).
Explanation
Understanding the Problem We are given a homomorphism from D 8 to S 4 defined by the action of the symmetries of a square on its vertices. We need to find the 8 permutations in the image of this homomorphism.
Identifying Elements of D8 Let's label the vertices of the square as 1, 2, 3, and 4 in clockwise order. The elements of D 8 are:
Identity (no change): e
Rotation by 90 degrees clockwise: r
Rotation by 180 degrees clockwise: r 2
Rotation by 270 degrees clockwise: r 3
Reflection about the horizontal axis (through vertices 1 and 3): h
Reflection about the vertical axis (through vertices 2 and 4): v
Reflection about the diagonal from vertex 1 to 4: d 1
Reflection about the diagonal from vertex 2 to 3: d 2
Finding Permutations in S4 Now, let's determine the corresponding permutations in S 4 for each element of D 8 :
Identity e : (1)(2)(3)(4) = ()
Rotation r (90 degrees): (1 2 3 4)
Rotation r 2 (180 degrees): (1 3)(2 4)
Rotation r 3 (270 degrees): (1 4 3 2)
Reflection h : (2 4)
Reflection v : (1 3)
Reflection d 1 : (2 3)
Reflection d 2 : (1 4)
Listing the Permutations Therefore, the 8 permutations in the image of the homomorphism D 8 → S 4 are:
(), (1 2 3 4), (1 3)(2 4), (1 4 3 2), (2 4), (1 3), (2 3), (1 4)
Examples
Understanding how symmetries of a square relate to permutations is fundamental in fields like crystallography, where the arrangement of atoms in a crystal lattice exhibits specific symmetries. By analyzing these symmetries using group theory, scientists can predict material properties and understand their behavior under different conditions. This connection between abstract algebra and real-world phenomena highlights the power of mathematical tools in solving practical problems.
The homomorphism from D 8 to S 4 translates the symmetries of a square into vertex permutations. The 8 permutations in the image of this homomorphism are: (), (1 2 3 4), (1 3)(2 4), (1 4 3 2), (2 4), (1 3), (2 3), and (1 4). Each permutation represents how the symmetries of the square map the vertices labeled 1, 2, 3, and 4.
;
