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).
We prove that Chevalley group over the field F of zero characteristic possess R ∞ property, if F has torsion group of automorphisms or F is an algebraically closed field which has finite transcendence degree over Q. As a consequence we obtain that the twisted conjugacy class [e] ϕ of unit element is a subgroup of Chevalley group if and only if ϕ is central automorphism.
In the present paper, we introduce the new construction of quandles. For a group G and its subset A we construct a quandle Q(G, A) which is called the (G, A)-quandle and study properties of this quandle. In particular, we prove that if Q is a quandle such that the natural map Q → G Q from Q to its enveloping group G Q is injective, then Q is the (G, A)-quandle for an appropriate group G and its subset A. Also we introduce the free product of quandles and study this construction for (G, A)-quandles. In addition, we classify all finite quandles with enveloping group Z 2 .
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