Classification of Finite Completions of a Class Locally D₈ Amalgams

Abstract

Let 𝒜 = (A ₁, A ₁₂, A ₂) be a locally D ₈ amalgam with a finite completion G. Suppose that A ₁ ∈ Syl ₂(G). We show that under these conditions |A ₁| ≤2⁵, or N _{A 1} (A ₁₂) is Abelian. As applications of our results, we determine all the finite completions G, up to O(G), in the case where N _{A 1} (A ₁₂) is non-Abelian.

Key Words:

• Amalgams

2000 Mathematics Subject Classification:

• Primary 20B25
• Secondary 05C25

ACKNOWLEDGMENT

We are grateful to the referee, whose comments have improved the article.

Notes

The authors are grateful to Sergey Shpectorov for suggesting this problem under this hypothesis.

Communicated by A. Turull.