Maths Blog: Subgroups and Centralizers

Algorithmic Verification of Algebraic Structures

Exploring Subgroups of Complex Numbers and Centralizers in Dihedral Groups

Abstract

This blog investigates two algebraic problems: verifying whether a constrained subset of complex numbers forms a subgroup under addition, and classifying centralizers in dihedral groups. We show that the set H = {a + bi | a, b ∈ ℝ, ab ≥ 0} fails closure and is not a subgroup of ℂ. We also provide a parity-dependent classification of centralizers in D_n, highlighting differences between odd and even n.

Problem 72: Subgroup of Complex Numbers

Consider H = {a + bi | a, b ∈ ℝ, ab ≥ 0} under addition. To be a subgroup, H must satisfy identity, closure, and inverse properties.

Identity

0 = 0 + 0i ∈ H since 0·0 = 0 ≥ 0. ✅

Closure

Let x = 2 + 0i ∈ H and y = 0 - 3i ∈ H. Then x+y = 2 - 3i. Here ab = 2·(-3) = -6 < 0, so 2 - 3i ∉ H. ❌

Counterexample: x = 2 + 0i, y = 0 - 3i ∈ H, but x+y = 2 - 3i ∉ H.

Conclusion

Since closure fails, H is not a subgroup of ℂ under addition.

Problem 78: Centralizers in Dihedral Group D_n

The dihedral group D_n consists of n rotations and n reflections. For an element g ∈ D_n, the centralizer C(g) = {x ∈ D_n | xg = gx}.

Case 1: n odd

• C(F) = {e, F} (only identity and the reflection itself)
• C(R) = {e, R, R², …, R^n-1} (all rotations)

Case 2: n even

• C(F) = {e, R^n/2, F, F·R^n/2} (identity, half-turn rotation, and two reflections)
• C(R) = {e, R, R², …, R^n-1} (all rotations)

Summary

• Odd n: reflections commute only with themselves and identity; rotations commute with all rotations.
• Even n: reflections through opposite vertices commute with half-turn rotation and related reflections; rotations commute with all rotations.

Conclusion

We proved that H fails subgroup criteria due to lack of closure, and we classified centralizers in D_n based on parity. These results illustrate how continuous and discrete algebraic structures can be rigorously verified and algorithmically formalized.