The twin group Tn is a Coxeter group generated by n − 1 involutions and the pure twin group P Tn is the kernel of the natural surjection of Tn onto the symmetric group on n letters. In this paper, we investigate structural aspects of twin and pure twin groups. We prove that the twin group Tn decomposes into a free product with amalgamation for n > 4. It is shown that the pure twin group P Tn is free for n = 3, 4, and not free for n ≥ 6. We determine a generating set for P Tn, and give an upper bound for its rank. We also construct a natural faithful representation of T4 into Aut(F7). In the end, we propose virtual and welded analogues of these groups and some directions for future work.

In this paper, a theory of quandle rings is proposed for quandles analogous to the classical theory of group rings for groups, and interconnections between quandles and associated quandle rings are explored.

Let $G$ be a group and $\varphi \in \Aut(G)$. Then the set $G$ equipped with the binary operation $a*b=\varphi(ab^{-1})b$ gives a quandle structure on $G$, denoted by $\Alex(G, \varphi)$ and called the generalised Alexander quandle. When $G$ is additive abelian and $\varphi = -\id_G$, then $\Alex(G, \varphi)$ is the well-known Takasaki quandle. In this paper, we determine the group of automorphisms and inner automorphisms of Takasaki quandles of abelian groups with no 2-torsion, and Alexander quandles of finite abelian groups with respect to fixed-point free automorphisms. As an application, we prove that if $G\cong (\mathbb{Z}/p \mathbb{Z})^n$ and $\varphi$ is multiplication by a non-trivial unit of $\mathbb{Z}/p \mathbb{Z}$, then $\Aut\big(\Alex(G, \varphi)\big)$ acts doubly transitively on $\Alex(G, \varphi)$. This generalises a recent result of \cite{Ferman} for quandles of prime order.Comment: 11 pp, minor typo correcte

In this paper we study different questions concerning automorphisms of quandles. For a conjugation quandle Q = Conj(G) of a group G we determine several subgroups of Aut(Q) and find necessary and sufficient conditions when these subgroups coincide with the whole group Aut(Q). In particular, we prove that Aut(Conj(G)) = Z(G) ⋊ Aut(G) if and only if either Z(G) = 1 or G is one of the groups Z2, Z 2 2 or Z3, what solves [3, Problem 4.8]. For a big list of Takasaki quandles T (G) of an abelian group G with 2-torsion we prove that the group of inner automorphisms Inn(T (G)) is a Coxeter group, what extends the result [3, Theorem 4.2] which describes Inn(T (G)) and Aut(T (G)) for an abelian group G without 2-torsion. We study automorphisms of certain extensions of quandles and determine some interesting subgroups of the automorphism groups of these quandles. Also we classify finite quandles Q with 3 ≤ k-transitive action of Aut(Q).

Let 1 → N → G → H → 1 be an abelian extension. The purpose of this paper is to study the problem of extending automorphisms of N and lifting automorphisms of H to certain automorphisms of G.