Problem 10 - Algebra: Pure and Applied

Question

In the dihedral group \(D_{4}\), find the normalizer of \(P=\\{e, \tau\\}\), where \(\tau\) is a flip. Is \(P\) a normal subgroup of \(D_{4}\) ?

Step-by-Step Solution

Answer

The normalizer of \(P\) is \(\{e, r^2, \tau\}\). \(P\) is not a normal subgroup of \(D_{4}\).

Step 1: Understand the Problem

The dihedral group \(D_{4}\) has elements corresponding to symmetries of a square, including rotations and flips. We are given a subgroup \(P = \{ e, \tau \}\) and need to find its normalizer in \(D_{4}\). The normalizer of a subgroup \(P\) in a group \(G\) is the largest subgroup of \(G\) in which \(P\) is normal.

Step 2: Determine Elements of \(D_{4}\)

\(D_{4}\) consists of 8 elements: identity \(e\), rotations \(r, r^2, r^3\) (each \(r\) represents a rotation of 90 degrees), and flips \(\tau, r\tau, r^2\tau, r^3\tau\).

Step 3: Examine Conjugation in \(D_{4}\)

We need to find which elements \(g\) in \(D_{4}\) satisfy \(gPg^{-1} = P\). This means examining how the conjugation action preserves \(P = \{e, \tau\}\).

Step 4: Conjugate Flip \(\tau\) by Rotations

Calculate conjugates: 1. \(r\tau r^{-1} = r\tau r^3 = r\tau r^2 = r^2\tau\), which is not in \(P\), 2. \(r^2\tau r^{-2} = \tau\), 3. \(r^3\tau r^3 = r^3\tau = r\tau\), which is not in \(P\). Thus, only \(r^2\) maintains \(P\).

Step 5: Check Flips for Normalization

Check all products of flips and \(\tau\): 1. \(\tau\tau\tau^{-1} = \tau\)2. Flip-conjugation by \(r\tau\) results in different identities not preserving \(P = \{e, \tau\}\).

Step 6: List Elements of the Normalizer

The elements that conjugate \(P\) to itself are \(e\), \(r^2\), and \(\tau\), forming the set \(N_{D_{4}}(P) = \{e, r^2, \tau\}\).

Step 7: Check Normality of \(P\) in \(D_{4}\)

A subgroup is normal in a group if it is equal to its normalizer in that group. Since \(N_{D_{4}}(P) = \{e, r^2, \tau\}\) and not the entire \(D_4\), \(P\) is not a normal subgroup of \(D_{4}\).

Key Concepts

Conjugation: A Symmetry Transformation

Conjugation in group theory is a way of transforming an element by another element of the group. It's like looking at an object from a different angle after a twist or flip. For any element \(g\) in a group \(G\), the conjugation of \(h\) by \(g\) is given by \(ghg^{-1}\). This operation is crucial in understanding how elements relate to each other. In our problem with the dihedral group \(D_4\), conjugation helps to determine which elements transform the subgroup \(P=\{e, \tau\}\) into itself.

• If \(gPg^{-1} = P\), then \(g\) is a part of the normalizer of \(P\).
• For instance, if we take \(r^2\) and conjugate \(\tau\), we find that \((r^2 \tau r^{2}) = \tau\), maintaining \(\tau\) in \(P\).

This preservation by conjugation indicates symmetry in the subgroup from certain group operations, such as specific rotations of the square that result in the same flips or identity position.

Normal Subgroup: Stability Under Conjugation

The concept of a normal subgroup is central to group theory. A subgroup \(H\) is considered normal in \(G\) if, for all elements \(g\) in \(G\), the conjugation \(gHg^{-1}\) results in \(H\) itself. Essentially, normal subgroups remain unchanged when symmetrically transformed by all elements of the entire group.In our exercise with \(D_4\), we determine whether \(P = \{e, \tau\}\) is normal by comparing it to its normalizer. If the normalizer \(N_G(P) = G\), then \(P\) is normal.

• Since \(N_{D_{4}}(P) = \{e, r^2, \tau\}\), which is not the entire \(D_4\), \(P\) is not a normal subgroup.

This tells us that some of the operations in \(D_4\) can transform \(P\) into something outside of itself, breaking the requirement for our subgroup to be normal.

Normalizer in Group Theory: The Largest Stable Umbrella

The normalizer of a subgroup \(H\) in a group \(G\) is an important concept that extends the idea of a normal subgroup. It identifies the largest part of \(G\) in which \(H\) acts like a normal subgroup. It is expressed as \(N_G(H)\) and is characterized by all \(g\) in \(G\) that keep \(H\) unchanged when conjugated.

• Mathematically, \(N_G(H) = \{ g \in G \mid gHg^{-1} = H \}\).

In the context of our dihedral group \(D_4\), the normalizer \(N_{D_4}(P)\) comprises elements \(e, r^2, \tau\) that preserve \(P = \{e, \tau\}\) under conjugation.The normalizer helps us understand the symmetry of the subgroup from within its environment, indicating how broadly a subgroup is well-behaved or invariant under these group operations. Here, the normalizer tells us that, beyond \(P\) itself, only specific symmetries like \(r^2\) maintain \(P\), indicating non-normalcy since it doesn’t span the whole group \(D_4\).