Dihedral Group D8

D8 is a group of symmetries of a square. It consists of eight elements, which can be represented by the rotations and reflections of the square.

Generators in D8

In the Dihedral group D8, the minimal number of generators refers to the minimum number of elements required to generate all the elements in the group. Let's analyze the generators in D8:

1. Rotations: The rotations in D8 consist of four elements, which are denoted by R0, R90, R180, and R270. Each rotation represents a clockwise rotation of the square by 0, 90, 180, and 270 degrees, respectively. These rotations form a cyclic subgroup of order 4.

2. Reflections: The reflections in D8 consist of four elements, denoted by F, V, D, and H. F represents a reflection along the vertical axis, V represents a reflection along the horizontal axis, D represents a reflection along the diagonal from the top left to the bottom right, and H represents a reflection along the diagonal from the top right to the bottom left. These reflections form a cyclic subgroup of order 2.

Minimal Number of Generators

To determine the minimal number of generators in D8, we need to find the smallest set of generators that can generate all eight elements of the group.

Since the rotations form a cyclic subgroup of order 4 and the reflections form a cyclic subgroup of order 2, we can choose any two elements, one from each subgroup, as generators. This is because the composition of a rotation and a reflection can generate all eight elements of D8.

Therefore, the minimal number of generators in D8 is two.

Example:

Let's choose R90 and F as the generators. By composing these two elements, we can generate all the elements of D8 as follows:
- R90 * F = D (reflection along the diagonal)
- R90^2 * F = V (reflection along the horizontal axis)
- R90^3 * F = H (reflection along the diagonal)
- R90^4 * F = R90 (rotation by 90 degrees)
- R90 * F^2 = R180 (rotation by 180 degrees)
- R90 * F * F = R270 (rotation by 270 degrees)
- R90^3 * F^2 = F (reflection along the vertical axis)
- R90^2 * F^2 = R0 (identity element)

Thus, with just R90 and F as generators, we can generate all eight elements of D8.

Summary:

The minimal number of generators in D8 is two. By choosing one rotation and one reflection as generators, we can generate all eight elements of the Dihedral group D8.