Question

Show that \(\operatorname{Aut}\left(D_{4}\right) \cong D_{4}\)

Short answer

The automorphism group of the dihedral group \(D_4\) is isomorphic to \(D_4\) itself.

Step-by-step solution

Understand the Problem

We need to show that the automorphism group of the dihedral group of order 8 (denoted as \(D_4\)) is isomorphic to \(D_4\) itself. An automorphism group of a group \(G\), denoted \(\operatorname{Aut}(G)\), is the group of all automorphisms of \(G\). Isomorphic groups have a one-to-one correspondence between their elements that preserves the group operation.

Define the Group \(D_4\)

The dihedral group \(D_4\) is the group of symmetries of a square, including both rotations and reflections. It has 8 elements: \(\{e, r, r^2, r^3, s, sr, sr^2, sr^3\}\), where \(r\) denotes a 90-degree clockwise rotation and \(s\) denotes a reflection.

Analyze the Group Structure

\(D_4\) is generated by two elements \(r\) and \(s\) with relations \(r^4 = e\), \(s^2 = e\), and \(srs = r^{-1}\). Any automorphism must map \(r\) to a generator of the cyclic subgroup \(\langle r \rangle\), meaning it can map to \(r\), \(r^3\), \(r^2\), or \(e\). \(s\) must map to an element whose square is \(e\).

Find the Order of \(\operatorname{Aut}(D_4)\)

The order of \(\operatorname{Aut}(G)\) for some group \(G\) is given by the number of automorphisms. We find each valid map of generators (\(r\) and \(s\)) that respects group relations. Using properties of \(D_4\), calculate distinct mappings to verify these match \(r\) and \(s\) mappings.

Verify Isomorphism

Show that the valid automorphisms can be paired with elements of \(D_4\) itself and contain the same group structure. Each element of \(\operatorname{Aut}(D_4)\) corresponds to exactly one element in \(D_4\) and respects group operations and generators' relations.

Key concepts

Dihedral group

A dihedral group, denoted as \(D_n\) for some positive integer \(n\), represents the group of symmetries of a regular \(n\)-sided polygon. The most common dihedral group people come across is \(D_4\), representing the symmetries of a square. It includes combinations of rotations and reflections that map the square onto itself without altering its shape.

For \(D_4\), there are 8 distinct elements representing:

• The identity transformation \(e\), doing nothing to the square.
• Three rotations: 90°, 180°, and 270° clockwise, denoted as \(r\), \(r^2\), and \(r^3\) respectively.
• Reflections about vertical, horizontal, and two diagonal axes, denoted as \(s\), \(sr\), \(sr^2\), and \(sr^3\).

These elements exhibit specific relationships like \(r^4 = e\), \(s^2 = e\), and composition rules like \(srs = r^{-1}\). Understanding \(D_4\) is the first step in discussing its automorphisms.

Group isomorphism

Group isomorphism is a significant concept that indicates a strong connection between two groups: they essentially have the same structure. Two groups \(G\) and \(H\) are said to be isomorphic, expressed \(G \cong H\), if there exists a bijective map (one-to-one and onto function) \(f: G \to H\) such that for any elements \(a\) and \(b\) in \(G\), \(f(ab) = f(a)f(b)\).

This means that not only are the sets \(G\) and \(H\) the same size, but their group operations correspond exactly. If you can show a bijection that preserves these operations, the groups are considered structurally identical, even if their elements or operations might look different at first glance.

Within the context of the dihedral group, showing that \(\operatorname{Aut}(D_4) \cong D_4\) would mean illustrating that all the symmetries as automorphisms have a corresponding structure like those of the original dihedral group.

Group automorphism

A group automorphism is a special type of isomorphism from a group to itself. This is not just any function, but a bijective map that preserves the group operation. For a group \(G\), an automorphism is a map \(\phi: G \to G\) such that \(\phi(ab) = \phi(a)\phi(b)\) for all \(a, b\) in \(G\), and there exists an inverse map ensuring every group element gets mapped bijectively.

In the case of \(D_4\), an automorphism would keep the relationships such as \(r^4 = e\) and \(s^2 = e\) true in the new mapping. Moreover, because \(\operatorname{Aut}(D_4)\) measures all such possible automorphisms, you need to explore all mappings of generators \(r\) and \(s\) that adhere to these rules. These mappings highlight how \(D_4\)'s structural properties are preserved even when viewed as distinct symmetries within itself.

Symmetries of a square

Understanding the symmetries of a square is crucial for grasping the properties of \(D_4\). A square has several symmetrical actions it can withstand which are realized as elements of \(D_4\).

These actions include:

• Rotation: A square can be rotated about its center by 90°, 180°, or 270° (clockwise or counterclockwise), and these rotations are reversible and form a rotation group within \(D_4\).
• Reflection: It can be reflected about an axis. The axes could be a vertical line through the center, a horizontal line through the center, or the two diagonals. Each reflection reverses orientation but results in an identical square layout.

These actions are not only standalone but can be composed, creating combined transformations that return the square into a previous symmetrical state like its initial position. These combined transformations manifest as elements such as \(sr\) or \(sr^2\). Exploring these symmetries deeply helps visualize how \(D_4\) works and serves as a foundation for identifying and classifying automorphisms.