Define a homomorphism $D_8 \rightarrow S_4$ by labeling vertices of a square, as we did for a triangle. List the 8 permutations in the image of this homomorphism.
Answer (1)
Label the vertices of the square as 1, 2, 3, 4.
Represent each element of D 8 as a permutation of the vertices.
List the permutations corresponding to the elements of D 8 .
The image of the homomorphism is: ( ) , ( 1234 ) , ( 13 ) ( 24 ) , ( 1432 ) , ( 12 ) ( 34 ) , ( 24 ) , ( 14 ) ( 23 ) , ( 13 ) .
Explanation
Understanding the Problem We are given a homomorphism from D 8 to S 4 defined by the action of D 8 on the vertices of a square. We need to find the image of this homomorphism, which consists of the 8 permutations in S 4 that correspond to the elements of D 8 .
Representing Elements of D8 as Permutations Let's label the vertices of the square as 1, 2, 3, and 4 in counterclockwise order. The elements of D 8 are the identity e , rotations r , r 2 , r 3 , and reflections s , sr , s r 2 , s r 3 , where r is a rotation by 90 degrees and s is a reflection about the vertical axis. We will represent each element of D 8 as a permutation of the vertices {1, 2, 3, 4}.
Identity Element
e (identity): This leaves all vertices unchanged, so the permutation is (1)(2)(3)(4), which can be written as ().
Rotation by 90 Degrees
r (rotation by 90 degrees): This maps 1 to 2, 2 to 3, 3 to 4, and 4 to 1. The permutation is (1 2 3 4).
Rotation by 180 Degrees
r 2 (rotation by 180 degrees): This maps 1 to 3, 2 to 4, 3 to 1, and 4 to 2. The permutation is (1 3)(2 4).
Rotation by 270 Degrees
r 3 (rotation by 270 degrees): This maps 1 to 4, 2 to 1, 3 to 2, and 4 to 3. The permutation is (1 4 3 2).
Reflection about Vertical Axis
s (reflection about the vertical axis): This swaps 1 and 2, and 3 and 4. The permutation is (1 2)(3 4).
Reflection about Diagonal 1-3
sr (reflection about the diagonal 1-3): This swaps 2 and 4. The permutation is (2 4).
Reflection about Horizontal Axis
s r 2 (reflection about the horizontal axis): This swaps 1 and 4, and 2 and 3. The permutation is (1 4)(2 3).
Reflection about Diagonal 2-4
s r 3 (reflection about the diagonal 2-4): This swaps 1 and 3. The permutation is (1 3).
Final Answer Therefore, the 8 permutations in the image of the homomorphism are: (), (1 2 3 4), (1 3)(2 4), (1 4 3 2), (1 2)(3 4), (2 4), (1 4)(2 3), (1 3).
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 and homomorphisms, scientists can predict material properties and understand crystal structures. For instance, the symmetry operations of a square lattice can be mapped to permutations of its vertices, aiding in the classification and analysis of 2D materials with square symmetry. This approach provides a powerful tool for studying and predicting the behavior of materials at the atomic level.
