For each pointed abelian group $(A,c)$, there is an associated 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 $m$.
@article{10_37236_2676,
author = {William E. Clark and Xiang-dong Hou},
title = {Galkin quandles, pointed abelian groups, and sequence {A000712}},
journal = {The electronic journal of combinatorics},
year = {2013},
volume = {20},
number = {1},
doi = {10.37236/2676},
zbl = {1267.05030},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2676/}
}
TY - JOUR
AU - William E. Clark
AU - Xiang-dong Hou
TI - Galkin quandles, pointed abelian groups, and sequence A000712
JO - The electronic journal of combinatorics
PY - 2013
VL - 20
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/2676/
DO - 10.37236/2676
ID - 10_37236_2676
ER -
%0 Journal Article
%A William E. Clark
%A Xiang-dong Hou
%T Galkin quandles, pointed abelian groups, and sequence A000712
%J The electronic journal of combinatorics
%D 2013
%V 20
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/2676/
%R 10.37236/2676
%F 10_37236_2676
William E. Clark; Xiang-dong Hou. Galkin quandles, pointed abelian groups, and sequence A000712. The electronic journal of combinatorics, Tome 20 (2013) no. 1. doi: 10.37236/2676
