Galkin Quandles, Pointed Abelian Groups, and Sequence $A000712$

TL;DR

This paper classifies all finite pointed abelian groups and explores their connection to Galkin quandles, which are algebraic structures used to construct knot invariants, revealing a combinatorial formula for their count.

Contribution

It provides a complete classification of finite pointed abelian groups and links this classification to Galkin quandles, including a formula for counting nonisomorphic groups.

Findings

Number of nonisomorphic pointed abelian groups of order q^n is sum of products of partition numbers.

Galkin quandles are uniquely determined by their associated pointed abelian groups.

The classification aids in understanding algebraic structures used in knot theory.

Abstract

For each pointed abelian group $(A,c)$, there is an associated {\em Galkin quandle} $G(A,c)$ which is an algebraic structure defined on $\Bbb Z_3\times A$ that can be used to construct knot invariants. It is known that two finite Galkin quandles are isomorphic if and only if their associated pointed abelian groups are isomorphic. In this paper we classify all finite pointed abelian groups. We show that the number of nonisomorphic pointed abelian groups of order $q^n$ ($q$ prime) is $\sum_{0\le m\le n}p(m)p(n-m)$, where $p(m)$ is the number of partitions of integer $m$.