[Paper Review] Embedding Properties in Central Products

Dandrielle Lewis, Almousa, Ayah+1 more|arXiv (Cornell University)|Aug 1, 2014

Finite Group Theory Research6 references3 citations

TL;DR

This paper characterizes normal, subnormal, and abnormal subgroups in central products of finite groups by leveraging commutator theory and group epimorphisms. It establishes equivalent conditions for these embedding properties, particularly proving that a subgroup is abnormal in a central product if and only if its preimage under the defining epimorphism is abnormal in the direct product, under solvability assumptions.

ABSTRACT

In this article, we study group theoretical embedding properties of subgroups in central products of finite groups. Specifically, we give characterizations of normal, subnormal, and abnormal subgroups of a central product of two groups.

Motivation & Objective

• To extend existing characterizations of subgroup embedding properties—normal, subnormal, and abnormal—from direct products to central products.
• To address the gap in the literature regarding subgroup structure in central products, especially for non-direct product constructions.
• To provide constructive criteria for identifying these subgroup types in central products using preimage analysis under epimorphisms.
• To lay foundational results for future work on pronormal subgroups and abnormal subgroups in non-solvable central products.

Proposed method

• Define central products both internally and externally, using an epimorphism ε: D → G where D = U₁ × U₂ and ker ε identifies the amalgamated center.
• Use commutator subgroups [A, B] to characterize normality in central products, showing H ◁ G iff [H, G] ≤ H.
• Apply the Lattice Isomorphism Theorem via epimorphism ε to relate subgroups of G to subgroups of D = U₁ × U₂.
• Prove that H is abnormal in G iff ε⁻¹(H) is abnormal in D, under the condition that at least one of U₁ or U₂ is solvable.
• Generalize results to subgroups H = V₁V₂ of G, showing H is abnormal in G iff each Vᵢ is abnormal in Uᵢ and ε⁻¹(H) = V₁ × V₂.
• Use standard group theory tools: commutator identities, normalizer conditions, and subgroup correspondence theorems.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a subgroup to be normal in a central product of two finite groups?
• RQ2 How can subnormal subgroups in central products be characterized using the derived series and subgroup chains?
• RQ3 Under what conditions is a subgroup H = V₁V₂ abnormal in a central product G = U₁U₂, where Vᵢ ≤ Uᵢ?
• RQ4 How does the abnormality of a subgroup in a central product relate to the abnormality of its preimage under the defining epimorphism?
• RQ5 Can the characterization of abnormal subgroups in central products be extended to cases where both factors are non-solvable?

Key findings

• A subgroup H of a central product G = U₁U₂ is normal if and only if the commutator [H, G] is contained in H.
• A subgroup H of G is subnormal if and only if its preimage under ε is subnormal in D = U₁ × U₂.
• H is abnormal in G if and only if ε⁻¹(H) is abnormal in D, provided that at least one of U₁ or U₂ is solvable.
• For H = V₁V₂ with Vᵢ ≤ Uᵢ and ε⁻¹(H) = V₁ × V₂, H is abnormal in G if and only if each Vᵢ is abnormal in Uᵢ.
• The characterization of abnormal subgroups in central products can be algorithmically implemented in computational algebra systems like GAP or Sage.
• The results generalize known characterizations from direct products to central products, extending tools from Goursat’s work to more complex group constructions.