Narrow Results

In this paper, we discuss the group consisting of some good automorphisms of representation ring of the non-pointed Hopf algebra $D(n)$, the quotient of the non-pointed prime Hopf algebras of GK-dimension one, which is generated by $x,y,z$ with the relations:

Let $s$ be a positive integer. A graph is $s$-transitive if its automorphism group is transitive on s-arcs but not on $(s+1)$-arcs. In this paper, we study all tetravalent $s$-transitive graphs of order $6p^2$.

The Bergman–Hartogs domain which can be regarded as a generalization of the Cartan–Hartogs domain provides a large class of bounded pseudoconvex domains which are in general nonhomogeneous. Since the geometry of a domain is determined by its automorphism group to a certain extent, it is meaningful to study the structure of the automorphism group. In this paper, we completely determine the structure of the holomorphic automorphism group of the Bergman–Hartogs domain over a minimal homogeneous domain with center at the origin.

The derivation algebra and automorphism group of a class of super $W$-algebra $W(2,2)$ are completely determined in this paper.

We classify finite groups acting by birational transformations of a nontrivial Severi–Brauer surface over a field of characteristic zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field $𝕂$ of characteristic zero that has no $𝕂$-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces.

We consider and resolve the gap problem for almost quaternion-Hermitian structures, i.e. we determine the maximal and submaximal symmetry dimensions, both for Lie algebras and Lie groups, in the class of almost quaternion-Hermitian manifolds. We classify all structures with such symmetry dimensions. Geometric properties of the submaximally symmetric spaces are studied, in particular, we identify locally conformally quaternion-Kähler structures as well as quaternion-Kähler with torsion.

Counterexamples to the $2$-jet determination Chern–Moser Theorem in codimension $d>2$ have recently been constructed [16]. We extend the Chern–Moser approach for hypersurfaces to real submanifolds of higher codimension in complex space to derive results on jet determination for their automorphism group. Using these techniques, we show that the $2$-jet determination Chern–Moser Theorem holds in codimension 2.

In this paper, we characterize weakly pseudoconvex domains of finite type in ${ℂ}^n$ in terms of the boundary behavior of automorphism orbits by using the scaling method.

The automorphism group of the spinor space of the standard model introduces the permutations of S₃ in addition to the gauge groups in a unified theory of the elementary particle interactions. The ${\mathrm{\text{S}}}_3$ symmetry will be the basis of a theoretical explanation of the inclusion of a fermion term, which transforms under the spin-$\frac{1}{2}$ representation of the symmetric permutation group, in the Koide relation for charged leptons. This field is identified with the sterile neutrino, and the mass is compatible with experimental bounds.

Denote by Φ₂ the automorphism group of the free group on two generators. We classify the indecomposable complex linear representations of Φ₂ of degree at most 5, and determine which ones have an infinite image. As a side-effect, our classification incorporates all the indecomposable representations of GL₂(Z) in these degrees.

Let be a free Burnside group of a sufficiently large odd exponent n with a basis of cardinality at least 2. We prove that every normal automorphism of is inner.

We also prove that a free Burnside group of large odd exponent n can be normally embedded into group of exponent n only as a direct factor.

A compact bordered Klein surface of algebraic genus p ≥ 2 has at most 12(p-1) automorphisms. Automorphism groups which attain this bound are called M*-groups. In this paper, firstly, we define generalized M*-groups. Then, we show that there is a relationship between the extended Hecke groups and generalized M*-groups. Finally, we prove that a generalized M*-groups G is supersoluble if and only if |G| = 4 · q^r for q ≥ 3 prime number and for some positive integer r.

We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following.

• If G is a nonelementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group.

• If H is a nonparabolic subgroup of a relatively hyperbolic group, and if any isometric H-action on an ℝ-tree is trivial, then H is Hopfian.

• If G is a nonelementary relatively hyperbolic group whose peripheral subgroups are finitely generated, then G has a nonelementary relatively hyperbolic quotient that is Hopfian.

• Any finitely presented group is isomorphic to a finite index subgroup of Out(H) for some group H with Kazhdan property (T). (This sharpens a result of Ollivier–Wise).

We show that the compressed word problem in a finitely generated fully residually free group (-group) is decidable in polynomial time, and use this result to show that the word problem in the automorphism group of an -group is decidable in polynomial time.

For any right-angled Artin group, we show that its outer automorphism group contains either a finite-index nilpotent subgroup or a nonabelian free subgroup. This is a weak Tits alternative theorem. We find a criterion on the defining graph that determines which case holds. We also consider some examples of solvable subgroups, including one that is not virtually nilpotent and is embedded in a non-obvious way.

We find a set of generators for the automorphism group Aut G of a graph product G of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup Aut^⋆ G defined in [Automorphisms of graph products of abelian groups, to appear in Groups, Geometry and Dynamics]. We follow closely the plan of M. Laurence's paper [A generating set for the automorphism group of a graph group, J. London Math. Soc. (2)52(2) (1995) 318–334].

In this paper, we describe the automorphism groups of the endomorphism semigroups of free Burnside groups B(m, n) for odd exponents n ≥ 1003. We prove, that the groups Aut(B(m, n)) and Aut(End(B(m, n))) are canonically isomorphic. In particular, if the groups Aut(End(B(m, n))) and Aut(End(B(k, n))) are isomorphic, then m = k.

The question of describing the automorphisms of $End(A)$ for a free algebra $A$ in a certain variety was considered by different authors since 2002. In this paper, we consider this question for the class of all relatively free groups having only cyclic centralizers of non-trivial elements. We prove that each automorphism of the endomorphism semigroup $End(F)$ of groups $F$ from this class is uniquely determined by its action on the subgroup of inner automorphisms $Inn(F)$. The obtained general result includes the following cases: absolutely free groups, free Burnside groups of odd period $n\ge 665$, free groups of infinitely based varieties of Adian (the cardinality of the set of such varieties is continuum), and so on.

We use the set symmetric difference between vertex subsets of a finite undirected simple graph $G$ to define a binary operation ∘ on the vertex set of a new graph $G^{\S }$, that contains $G$ as subgraph and whose vertices are non-empty vertex subsets of $G$. We show how the binary operation ∘ determines an algebraic structure on $V(G^{\S })$ that is strictly related to the graph structure of $G^{\S }$. In fact, we show that $\mathrm{\text{Aut}}(V(G^{\S }),\circ )$ agrees with $\mathrm{\text{Aut}}(G^{\S })$ and, next, we provide several characterizations for the algebraic structure $(V(G^{\S }),\circ )$ when the graph $G^{\S }$ is connected and locally dissymmetric.

We consider the Macdonald group $〈x,y|x^{[x,y]}=x^{1+2^m\ell },y^{[y,x]}=y^{1+2^m\ell }〉$ and its Sylow 2-subgroup $J=〈x,y|x^{[x,y]}=x^{1+2^m\ell },y^{[y,x]}=y^{1+2^m\ell },x^{2^{3m-1}}=y^{2^{3m-1}}=1〉$, where $m\ge 1$ and $\ell$ is odd. Then J has order $2^{7m-3}$, and nilpotency class 5 if $m>1$ and 3 if $m=1$. We determine the automorphism group of the 2-groups J, $H=J/Z(J)$ and $K=H/Z(H)$, where $\left|H\right|=2^{6m-3}$ and $\left|K\right|=2^{5m-3}$. Explicit multiplication, power, and commutator formulas for J, H and K are given, and used in the calculation of $\mathrm{\text{Aut}}(J)$, $\mathrm{\text{Aut}}(H)$ and $\mathrm{\text{Aut}}(K)$.