Embeddings of quandles into groups

Abstract

In this paper, we introduce the new construction of quandles. For a group [Formula: see text] and a subset [Formula: see text] of [Formula: see text] we construct a quandle [Formula: see text] which is called the [Formula: see text]-quandle and study properties of this quandle. In particular, we prove that if [Formula: see text] is a quandle such that the natural map [Formula: see text] from [Formula: see text] to the enveloping group [Formula: see text] of [Formula: see text] is injective, then [Formula: see text] is the [Formula: see text]-quandle for an appropriate group [Formula: see text] and a subset [Formula: see text] of [Formula: see text]. Also we introduce the free product of quandles and study this construction for [Formula: see text]-quandles. In addition, we classify all finite quandles with enveloping group [Formula: see text].