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 investigate free actions of some compact groups on cohomology real and complex Milnor manifolds. More precisely, we compute the mod 2 cohomology algebra of the orbit space of an arbitrary free ℤ2 and $\mathbb{S}^1$-action on a compact Hausdorff space with mod 2 cohomology algebra of a real or a complex Milnor manifold. As applications, we deduce some Borsuk–Ulam type results for equivariant maps between spheres and these spaces. For the complex case, we obtain a lower bound on the Schwarz genus, which further establishes the existence of coincidence points for maps to the Euclidean plane.
The Dold manifold P (m, n) is the quotient of S m × CP n by the free involution that acts antipodally on S m and by complex conjugation on CP n . In this paper, we investigate free actions of finite groups on products of Dold manifolds. We show that if a finite group G acts freely and mod 2 cohomologically trivially on a finite-dimensional CW-complex homotopy equivalent to k i=1 P (2mi, ni), then G ∼ = (Z2) l for some l ≤ k. This is achieved by first proving a similar assertion for k i=1 S 2m i × CP n i . We also determine the possible mod 2 cohomology algebra of orbit spaces of arbitrary free involutions on Dold Manifolds, and give an application to Z2-equivariant maps.
Let [Formula: see text] be a group and [Formula: see text] be an automorphism of [Formula: see text]. Two elements [Formula: see text] are said to be [Formula: see text]-twisted conjugate if [Formula: see text] for some [Formula: see text]. We say that a group [Formula: see text] has the [Formula: see text]-property if the number of [Formula: see text]-twisted conjugacy classes is infinite for every automorphism [Formula: see text] of [Formula: see text]. In this paper, we prove that twisted Chevalley groups over a field [Formula: see text] of characteristic zero have the [Formula: see text]-property as well as the [Formula: see text]-property if [Formula: see text] has finite transcendence degree over [Formula: see text] or [Formula: see text] is periodic.
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