Group Aut Research Articles

On the number of solutions of some Frobenius type equations in finite groups

Let G be a finite group with automorphism group Aut ( G ) and g ∈ G . For any subgroup H of Aut ( G ) , define a map ξ H : G → C by ξ H ( g ) = | { ( x , α ) ∈ G × H : [ x , α ] = g } | , where [ x , α ] : = x − 1 α ( x ) is called the autocommutator of x and α in G. In this paper, we show that ξ H is a character of G when H is any subgroup of Aut ( G ) containing Inn ( G ) and H = Autcent ( G ) , where Inn ( G ) is the inner automorphism group of G and Autcent ( G ) is the group of all the central automorphisms of G. In these cases expressions for ξ H are also obtained in terms of irreducible characters of G. Our results generalize a classical result of Frobenius regarding the number of solutions of the commutator equation in finite groups. As applications of our results, we compute g-autocommuting probabilities of some finite groups and obtained certain bounds for g-autocommuting probability of G in terms of commuting and autocommuting probabilities.

Realizing regular representations of finite groups

Given a regular representation of a finite group G and a positive integer number n, we construct a (finite) topological space X such that its group of homotopy classes of self-homotopy equivalences E(X) and its group of homeomorphisms Aut(X) are isomorphic to G, and the action of G on the n-th homology group Hn(X) is the regular representation. We also discuss other representations.

New upper bounds on the number of non-zero weights of constacyclic codes

For any simple-root constacyclic code C over a finite field Fq, as far as we know, the group G generated by the multiplier, the constacyclic shift and the scalar multiplications is the largest subgroup of the automorphism group Aut(C) of C. In this paper, by calculating the number of G-orbits of C﹨{0}, we give an explicit upper bound on the number of non-zero weights of C and present a necessary and sufficient condition for C to meet the upper bound. Some examples in this paper show that our upper bound is tight and better than the upper bounds in Zhang and Cao (2024) [26]. In particular, our main results provide a new method to construct few-weight constacyclic codes. Furthermore, for the constacyclic code C belonging to two special types, we obtain a smaller upper bound on the number of non-zero weights of C by substituting G with a larger subgroup of Aut(C). The results derived in this paper generalize the main results in Chen et al. (2024) [9].

On normal subgroups in automorphism groups

Abstract We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) . In particular, we prove that a finite normal subgroup in Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) has at most order two and if Γ is not a clique, then any finite normal subgroup in Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) is trivial. This property has implications for automatic continuity and C ∗ C^{\ast} -algebras: every algebraic epimorphism φ : L ↠ Aut ⁢ ( A Γ ) \varphi\colon L\twoheadrightarrow\mathrm{Aut}(A_{\Gamma}) from a locally compact Hausdorff group 𝐿 is continuous if and only if A Γ A_{\Gamma} is not isomorphic to Z n \mathbb{Z}^{n} for any n ≥ 1 n\geq 1 . Furthermore, if Γ is not a join and contains at least two vertices, then the set of invertible elements is dense in the reduced group C ∗ C^{\ast} -algebra of Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) . We obtain similar results for Aut ⁢ ( G Γ ) \mathrm{Aut}(G_{\Gamma}) , where G Γ G_{\Gamma} is a graph product of cyclic groups. Moreover, we give a description of the center of Aut ⁢ ( G Γ ) \mathrm{Aut}(G_{\Gamma}) in terms of the defining graph Γ.

Associations between autistic traits, depression, social anxiety and social rejection in autistic and non-autistic adults

Autistic people frequently experience negative judgements from non-autistic people, often fuelled by misconceptions that autistic people lack empathy. Understanding responses to negative social judgement among autistic people is crucial because of the potential negative impact on wellbeing and future interactions. We investigated the role of autistic traits, social anxiety, and depression on behavioural indices of social rejection in 20 autistic (AUT; 11 males) and 40 non-autistic (N-AUT; 21 males) university students. Participants completed the Social Judgement Task (SJT) where they predicted whether they were liked by another person, then received feedback on whether those evaluations were correct. Participants also completed an Age Judgement Task (AJT) where they estimated the age of the pictured person. The AUT group had lower positive expectation scores, meaning less tendency to predict being liked. Across the whole sample, higher social anxiety predicted greater tendency to anticipate rejection from others, not autistic traits. These findings suggest early experiences of rejection might lead to a negative self-bias in autistic people and emphasise the importance of using a transdiagnostic approach by showing that social anxiety rather than autistic traits is associated with expectation of social rejection.

Radiant toric varieties and unipotent group actions

We consider complete toric varieties X such that a maximal unipotent subgroup U of the automorphism group Aut(X) acts on X with an open orbit. It turns out that such varieties can be characterized by several remarkable properties. We study the set of Demazure roots of the corresponding complete fan, describe the structure of a maximal unipotent subgroup U in Aut(X), and find all regular subgroups in U that act on X with an open orbit.

On the automorphisms of the Drinfel'd double of a Borel Lie subalgebra

Let g be a complex simple Lie algebra with a Borel subalgebra b. Consider the semidirect product Ib=b⋉b⁎, where the dual b⁎ of b is equipped with the coadjoint action of b and is considered as an abelian ideal of Ib. We describe the automorphism group Aut(Ib) of the Lie algebra Ib. In particular we prove that it contains the automorphism group of the extended Dynkin diagram of g. In type An, the dihedral subgroup was recently proved to be contained in Aut(Ib) by Dror Bar-Natan and Roland van der Veen in [1] (where Ib is denoted by Iun). Their construction is ad hoc and they asked for an explanation which is provided by this note. Let n denote the nilpotent radical of b. We obtain similar results for Ib‾=b⋉n⁎ that is both an Inönü-Wigner contraction of g and the quotient of Ib by its center.

Neighbour-transitive codes in Kneser graphs

A code C is a subset of the vertex set of a graph and C is s-neighbour-transitive if its automorphism group Aut(C) acts transitively on each of the first s+1 parts C0,C1,…,Cs of the distance partition{C=C0,C1,…,Cρ}, where ρ is the covering radius of C. While codes have traditionally been studied in the Hamming and Johnson graphs, we consider here codes in the Kneser graphs. Let Ω be the underlying set on which the Kneser graph K(n,k) is defined. Our first main result says that if C is a 2-neighbour-transitive code in K(n,k) such that C has minimum distance at least 5, then n=2k+1 (i.e., C is a code in an odd graph) and C lies in a particular infinite family or is one particular sporadic example. We then prove several results when C is a neighbour-transitive code in the Kneser graph K(n,k). First, if Aut(C) acts intransitively on Ω we characterise C in terms of certain parameters. We then assume that Aut(C) acts transitively on Ω, first proving that if C has minimum distance at least 3 then either K(n,k) is an odd graph or Aut(C) has a 2-homogeneous (and hence primitive) action on Ω. We then assume that C is a code in an odd graph and Aut(C) acts imprimitively on Ω and characterise C in terms of certain parameters. We give examples in each of these cases and pose several open problems.

A Graph with a Locally Projective Vertex-Transitive Group of Automorphisms Aut($$Fi_{22}$$) Which Has a Nontrivial Stabilizer of a Ball of Radius $$2$$

A Graph with a Locally Projective Vertex-Transitive Group of Automorphisms Aut($$Fi_{22}$$) Which Has a Nontrivial Stabilizer of a Ball of Radius $$2$$

Some two-weight codes invariant under the 3-fold covers of the Mathieu groups M22 and Aut(M22)

Using an approach from finite group representation theory we construct quaternary non-projective codes with parameters [Formula: see text], quaternary projective codes with parameters [Formula: see text] and [Formula: see text] and binary projective codes with parameters [Formula: see text] as examples of two-weight codes on which a finite almost quasisimple group of sporadic type acts transitively as permutation groups of automorphisms. In particular, we show that these codes are invariant under the [Formula: see text]-fold covers [Formula: see text] and [Formula: see text], respectively, of the Mathieu groups [Formula: see text] and [Formula: see text]. Employing a known construction of strongly regular graphs from projective two-weight codes we obtain from the binary projective (respectively, quaternary projective) two-weight codes with parameters those given above, the strongly regular graphs with parameters [Formula: see text] and [Formula: see text] respectively. The latter graph can be viewed as a [Formula: see text]-[Formula: see text]-symmetric design with the symmetric difference property whose residual and derived designs with respect to a block give rise to binary self-complementary codes meeting the Grey–Rankin bound with equality.

Joint Engagement in Mother-Child Dyads of Autistic and Non-Autistic Children Among Asian Indian Tamil Speaking Families.

This study profiled various levels of engagement and related communication behaviours among 50 Asian Indian Tamil autistic children(AUT) and their mothers. The interaction was compared with two groups of mother-child dyads of non-autistic (NA) children, 50 in each group, matched for chronological age (CA), and for language level (LL). Results indicated that despite mother's efforts to engage with their children, autistic children were often 'engaged with objects' or remained 'unengaged' due to children's preference for solitary play, while NA children were often engaged in 'co-ordinated' and 'people engagement'. Across the three groups, mothers predominantly took the lead and dominated the interaction, irrespective of children's language levels. These initiations by the mothers were often to provide instructions and to ask 'What' questions. Autistic children initiated communication predominantly to ask for an object and responded often in the form of negations and protests with limited verbal output or non-verbally. Most of the communication behaviours of both children and mothers in AUT group was quantitatively and qualitatively different when compared to those in both the NA groups, indicating unique nature of interactions despite matching for CA or LL. The observations from the study highlights the need for considering adult's contingent behaviours also, while assessing communication skills of autistic children in order to provide effective intervention.

Invariant spaces of holomorphic functions on the Siegel upper half-space

In this paper we consider the (ray) representations of the group Aut of biholomorphisms of the Siegel upper half-space U defined by Us(φ)f=(f∘φ−1)(Jφ−1)s/2, s∈R, and characterize the semi-Hilbert spaces H of holomorphic functions on U satisfying the following assumptions:(a)H is strongly decent;(b)Us induces a bounded ray representation of the group Aff of affine automorphisms of U in H. We use this description to improve the known characterization of the semi-Hilbert spaces of holomorphic functions on U satisfying (a) and (b) with Aff replaced by Aut.In addition, we characterize the mean-periodic holomorphic functions on U under the representation U0 of Aff.

Full automorphism groups of large order of compact bordered Klein surfaces

Let S be a compact bordered Klein surface of algebraic genus g≥2, and Aut(S) its full group of automorphisms, which is known to have order at most 12(g−1). In this paper we consider groups G of automorphisms of order at least 4(g−1) acting on such surfaces, and study whether G is the full group Aut(S) or, on the contrary, the action of G extends to a larger group. The extendability of the action depends first on the NEC signature with which G acts and, in some cases, also on whether a monodromy presentation of G admits or not a particular automorphism. For each signature we study which of the three possibilities [Aut(S):G]=1, 2 or 3 occur, and show that, whenever a possibility occurs, it occurs for infinitely many values of g. We find infinite families of groups G, explicitly described by generators and relations, which satisfy the corresponding equality.

On the structure group of an infinite dimensional JB-algebra

We extend several results for the structure group of a real Jordan algebra V, to the setting of infinite dimensional JB-algebras. We prove that the structure group Str(V), the cone preserving group G(Ω) and the automorphism group Aut(V) of the algebra V are embedded Banach-Lie groups of GL(V), and that each of the inclusions Aut(V)⊂G(Ω)⊂Str(V) are of embedded Banach-Lie subgroups. We give a full description of the components of Str(V) via cones, isotopes and central projections. We apply these results to V=B(H)sa the special JB-algebra of self-adjoint operators on an infinite dimensional complex Hilbert space, describing the groups Str(V),G(Ω),Aut(V), their Banach-Lie algebras and their connected components. We show that the action of the unitary group of H on Aut(V) has smooth local cross sections, thus Aut(V) is a smooth principal bundle over the unitary group, with structure group S1.

Automorphism Groups of Regular Elements with Von-Neumann Inverses of Local Near-Rings Admitting Frobenius Derivations

This paper presents the classication of the invariant subgroups of the automorphism groups of the regular elements obtained from nite local near-rings, the appropriate algebraic structure to study non-linear functions on finite groups. Just as rings of matrices operate on vector spaces, near-rings operate on groups. In this paper, we construct classes of zero symmetric local near-ring of characteristic pk; k = 1; 2 ; k \(\ge\) 3 admitting frobenius derivations, characterize the structures of the cyclic groups generated by the regular elements R(N) as well as the structures and the orders of the automorphism groups Aut(R(N)) of the regular elements.

The A-Möbius function of a finite group

The Möbius function of the subgroup lattice of a finite group G has been introduced by Hall and applied to investigate several different questions. We propose the following generalization. Let A be a subgroup of the automorphism group Aut(G) of a finite group G and denote by CA(G) the set of A-conjugacy classes of subgroups of G. For H ≤ G let [H]A = { Ha ∣ a∈ A} be the element of CA(G) containing H. We may define an ordering in CA(G) in the following way: [H]A ≤ [K]A if Ha ≤ K for some a ∈ A. We consider the Möbius function μA of the corresponding poset and analyse its properties and possible applications.

Regular Elements and Von-Neumann Inverses of a Class of Zero Symmetric Local Near-Rings Admitting Frobenius Derivations

Let N be a zero-symmetric local near-ring. An element \({x}\) \(\in\) N is either regular, zero or a zero divisor. In this paper, we construct a class of zero symmetric local near-ring of characteristic pk; k \(\ge\) 3 admitting an identity frobenius derivation, characterize the structures and orders of the set R(N), the regular compartment with an aim of advancing the classication problem of algebraic structures. The number theoretic notions relating the number of regular elements to Euler's phi-function and the arithmetic functions of Galois near-rings are adopted. Using the Fundamental Theorem of nitely generated Abelian groups, the structures of R(N) are proved to be isomorphic to cyclic groups of various orders. The study also extends to the automorphism groups Aut(R(N)) of the regular elements.

Four-Dimensional Real Division Algebras with Few Derivations

In order to advance in the determination of all four-dimensional real division algebras 𝒜, we introduce a new duplication process preserving the unit. This duplication process accompanied by an isotopy allow us to obtain all of these algebras in case Der(𝒜) = {0} and partially in case Der(𝒜) = {0}. In the last case, we provide an 8-parameter family of ugly ℂ-associative algebras and an 8-parameter family of ℂ-associative algebras whose group of automorphisms contains only the identity and some reflections. For non-ugly algebras ℍf,f, the group Aut(ℍf,f) contains a reflection. Also algebras 𝒜 with Aut(𝒜) = ℤ2 or ℤ2 × ℤ2 are studied.

On the group of automorphisms of the semigroup BFZ with the family F of inductive nonempty subsets of ω

We study automorphisms of the semigroup BFZ with the family F of inductive nonempty subsets of ω and provethat the group Aut(BFZ) of automorphisms of the semigroup BFZ is isomorphic to the additive group of integers.

On the liftability of the automorphism group of smooth hypersurfaces of the projective space

Let X be a smooth hypersurface of dimension n ≥ 1 and degree d ≥ 3 in the projective space given as the zero set of a homogeneous form F. If (n, d) ≠ (1, 3), (2, 4) it is well known that every automorphism of X extends to an automorphism of the projective space, i.e., Aut(X) ⊆ PGL(n + 2, ℂ). We say that the automorphism group Aut(X) is liftable if there exists a subgroup of GL(n + 2, ℂ) projecting isomorphically onto Aut(X) and leaving F invariant. Our main result in this paper shows that the automorphism group of every smooth hypersurface of dimension n and degree d is liftable if and only if d and n + 2 are relatively prime. We also provide an effective criterion to compute all the integers which are a power of a prime number and that appear as the order of an automorphism of a smooth hypersurface of dimension n and degree d. As an application, we give a sufficient condition under which some Sylow p-subgroups of Aut(X) are trivial or cyclic of order p.