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Given a construction f on groups, we say that a group G is f -realisable if there is a group H such that G ≅ f (H) , and completely f-realisable if there is a group H such that G ≅ f (H) and every subgroup of G is isomorphic to f (H 1) for some subgroup H 1 of H and vice versa. Denote by L (G) the absolute center of a group G, that is the set of elements of G fixed by all automorphisms of G. By using the structure of the automorphism group of a ZM-group, in this paper we prove that cyclic groups C N , N ∈ N * , are completely L-realisable. [ABSTRACT FROM AUTHOR]

In this paper, we consider the orderm k-automorphisms ofSL(2,k). We first characterize the forms that orderm k-automorphisms ofSL(2,k) take and then we find simple conditions on matricesAandB, involving eigenvalues and the field that the entries ofAandBlie in, that are equivalent to isomorphy between the orderm k-automorphismsInnAandInnB. We examine the number of isomorphy classes and conclude with examples for selected fields. [ABSTRACT FROM PUBLISHER]

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined. [ABSTRACT FROM PUBLISHER]

An automorphismαof a groupGis called aweakly power automorphismif it maps every non-periodic subgroup ofGonto itself. The aim of this paper is to investigate the behavior of weakly power automorphisms. In particular, among other results, it is proved that all weakly power automorphisms of a soluble non-periodic groupGof derived length at most 3 are power automorphisms, i.e. they fix all subgroups ofG. This result is best possible, as there exists a soluble non-periodic group of derived length 4 admitting a weakly power automorphism, which is not a power automorphism. [ABSTRACT FROM PUBLISHER]

Let be a prime integer and let be a free metabelian group of rank In the paper an automorphism of order permuting cyclically fixed bases of is studied. The subgroup of fixed points of and the properties of action of on the commutator subgroup of are described. There are also given some results concerning such automorphisms of free abelian groups of rank [ABSTRACT FROM AUTHOR]

Let be a noncommutative prime ring and an automorphism of , . We denote by the additive subgroup of generated by the subset and by the subring of generated by . In this paper we prove that if , then contains for some nonzero ideal of . Also, we prove the following theorem concerning : Suppose that . Then contains a nonzero ideal of if and only if either or . [ABSTRACT FROM AUTHOR]

The purpose of this paper is to classify the normal extensions G of a cyclic p-group C[sub n] i.e., G ... C[sub n], with the irreducible character induced from a faithful linear character of C[sub n]. In fact we determine all normal extensions G of a cyclic p-group C[sub n] by p-subgroups of the automorphism group Aut(C[sub n]) with the centralizer C[sub G](C[sub n]) = C[sub n]. Consequently, we have a theorem (Theorem 1) which might characterize the dihedral groups and the generalized quaternion groups of order 2[sup n] among non-abelian 2-groups. [ABSTRACT FROM AUTHOR]

In this paper we make an attempt to study right loops (S,o) in which, for eachy ∈ S, the map σyfrom the inner mapping groupGSof (S,o) to itself given by σy(h)(x)o h(y) = h(xoy),x ∈ S,h ∈ GSis a homomorphism. The concept of twisted automorphisms of a right loop and also the concept of twisted right gyrogroup appears naturally and it turns out that the study is almost equivalent to the study of twisted automorphisms and a twisted right gyrogroup. We also study relationship between twisted gyrotransversals and twisted subgroups. [ABSTRACT FROM PUBLISHER]

We call the action of an automorphism α of a finite groupGa Hughes type action if it is described by conditions on the orders of elements ofG ⟨ α ⟩ − G. In the present paper we study the structure of finite groupGadmitting an automorphism α of prime orderpso that the orders of elements inG ⟨ α ⟩ − Gare not divisible byp2. [ABSTRACT FROM AUTHOR]

LetGbe a group. By using a family 𝒜 of subsets of automorphisms ofG, we introduced a simple graph Γ𝒜(G), which is a generalization of the non-commuting graph. In this paper, we study the combinatorial properties of Γ𝒜(G). [ABSTRACT FROM AUTHOR]

A power automorphism θ of a group G is said to be pre-fixed-point-free if CG(θ) is an elementary abelian 2-group. G is called an E-group if G has a pre-fixed-point-free power automorphism. In this paper, finite E-groups, together with all their pre-fixed-point-free power automorphisms, are completely determined. Moreover, a characteristic of finite abelian groups is given, which explains some known facts concerning power automorphisms. [ABSTRACT FROM PUBLISHER]

The test rank tr( G ) of G is the minimum cardinality of a test set. In this paper we prove: I. Let N = N rc be a free nilpotent group of rank r = 2 and class c = 2. Then (i) tr( N ) = 2 for r odd and c = 2; (ii) tr( N ) = 1 in all other cases; (iii) an element g ? N 2 q , 2 is a test element if and only if it can be written as g = · s , where t 1 ,..., t q are nonzero integers. II. Let F r be a free group of rank r = 2 and A n be a free abelian group of rank n = 1. Then the group G = G rn = F r × A n has the test rank n . [ABSTRACT FROM AUTHOR]

In this paper we explicitly determine the derivation algebra, automorphism group of quasi Ln-filiform Lie algebras, and applying some properties of root vector decomposition we obtain an isomorphism theorem of quasi Ln-filiform Lie algebras. [ABSTRACT FROM AUTHOR]

The main result of this paper is an algorithm which determines the Kac parameters for every Z-gradation of the type &stilde; of classical affine Lie algebras, i.e., Lie algebras with Dynkin diagrams of the type Xl(1) = Al(1),Bl(1),Cl(1),Dl(1). We would like to emphasize that the algorithm determines also the corresponding basis of the root system of the classical affine Lie algebra. Such a basis we name the Kac basis. [ABSTRACT FROM AUTHOR]

A subgroup of a group G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be S-quasinormally embedded in G if every Sylow subgroup of H is a Sylow subgroup of some S-quasinormal subgroup of G. In this paper we examine the structure of a finite group G under the assumption that certain abelian subgroups of prime power order are S-quasinormally embedded in G. Our results improve and extend recent results of Ramadan [Ramadan, M. (2001). The influence of S-quasinormality of some subgroups of prime power order on the structure of finite groups. Arch. Math. 77:143-148]. [ABSTRACT FROM AUTHOR]

In this paper we study the relationship between some homological properties such as the weak and the global dimension, the strong n-coherence, and the (n, d)-property, where n and d are two integers, of a commutative ring and its subrings retract. A special application is the transfer of these properties from a commutative ring to its fixed subring with respect to a subgroup of its group of automorphisms. It concludes with a discussion of the scopes and limits of our results. [ABSTRACT FROM AUTHOR]

Let G a simple group of type 2B2(q) or 2G2(q), where q is an odd power of 2 or 3, respectively. The main goal of this paper is to determine the multiplicity free permutation representations of G and A ≤ Aut(G) where A is a subgroup containing a copy of G. Let B be a Borel subgroup of G. If G = 2B2(q) we show that there is only one non-trivial multiplicity free permutation representation, namely the representation of G associated to the action on G/B. If G = 2G2(q) we show that there are exactly two such non-trivial representations, namely the representations of G associated to the action on G/B and the action on G/M, where M = UC with U the maximal unipotent subgroup of B and C the unique subgroup of index 2 in the maximal split torus of B. The multiplicity free permutation representations of A correspond to the actions on A/H where H is isomorphic to a subgroup containing B if G = 2B2(q), and containing M if G = 2G2(q). The problem of determining the multiplicity free representations of the finite simple groups is important, for example, in the classification of distance-transitive graphs. [ABSTRACT FROM AUTHOR]

Let G = C p n be a finite cyclic p-group, and let Hol (G) denote its holomorph. In this work, we find and characterize the regular subgroups of Hol (G) that mutually normalize each other in the symmetric group Sym (G) . We represent regular subgroups of Hol (G) as vertices of a graph, and we connect a pair of them by an edge when they mutually normalize each other. The approach to construct this local normalizing graph relies on the theory of gamma functions, and the final result will contain all the information about the regular subgroups of Hol (G) in a compact form. Communicated by Alberto Facchini [ABSTRACT FROM AUTHOR]

Extending the notion of probability to the automorphisms of a group, we find the probability of an arbitrarily chosen automorphism of a group fixing an arbitrary element of the group. [ABSTRACT FROM PUBLISHER]

We prove that all automorphisms of the automorphism group of an infinitely generated free abelian group are inner. [ABSTRACT FROM PUBLISHER]

We find a remarkable relation between the Nagata type polynomial automorphisms and rectifiable space lines of a certain form. [ABSTRACT FROM AUTHOR]

The maximality of certain symplectic subgroups of unitary groups PSUn(K), n ≥ 4, (K any field admitting a non-trivial involutory automorphism) belonging to the class c5 of Aschbacher is proved. Furthermore some related geometry in the case n = 4 and K finite is investigated. [ABSTRACT FROM AUTHOR]

Applies computational methods to determine the efficiency of simple groups. Presentations for the minimal generating pairs for such groups; Automorphisms.

In this article we defineG-algebras, that is, graded algebras on which a reductive groupG, acts as gradation preserving automorphisms. Starting from a finite dimensionalG-moduleVand the polynomial ringℂ[V], it is shown how one constructs a sequence of projective varietiesVksuch that each point ofVkcorresponds to a graded algebra with the same decomposition up to degreekas aG-module. After some general theory, we apply this to the case thatVis then+1-dimensional permutation representation ofSn+1, the permutation group onn+1 letters. [ABSTRACT FROM AUTHOR]

Automorphic loops are loops in which all inner mappings are automorphisms. A large class of automorphic loops is obtained as follows: Letmbe a positive even integer,Gan abelian group, and α an automorphism ofGthat satisfies α2 = 1 ifm > 2. Then thedihedral-like automorphic loopDih(m,G, α) is defined on ℤm × Gby We prove that two finite dihedral-like automorphic loops Dih(m,G, α),are isomorphic if and only if,, and α is conjugate toin the automorphism group ofG. Moreover, for a finite dihedral-like automorphic loopQ, we describe the structure of the automorphism group ofQand its subgroup consisting of inner mappings ofQ. [ABSTRACT FROM AUTHOR]

We prove that the tame automorphism group TAut(Mn) of a free metabelian Lie algebraMninnvariables over a fieldkis generated by a single nonlinear automorphism modulo all linear automorphisms ifn ≥ 4 except the case whenn = 4 and char(k) ≠ 3. If char(k) = 3, then TAut(M4) is generated by two automorphisms modulo all linear automorphisms. We also prove that the tame automorphism group TAut(M3) cannot be generated by any finite number of automorphisms modulo all linear automorphisms. [ABSTRACT FROM AUTHOR]

LetFbe a relatively free algebra of infinite rank ϰ. We say thatFhas thesmall index propertyif any subgroup of Γ = Aut(F) of index at most ϰ contains the pointwise stabilizer Γ(U)of a subsetUofFof cardinality less than ϰ. We prove that every infinitely generated free nilpotent/abelian group has the small index property, and discuss a number of applications. [ABSTRACT FROM AUTHOR]

We compute the groups Aut(G) and Out(G) whereGis a crystallographic group of rank 2, or so-called wallpaper group. [ABSTRACT FROM AUTHOR]

Let r, s be two parameters chosen from ℂ* such that rmsn = 1 implies m = n = 0. We compute the derivations of the two-parameter quantized enveloping algebraand calculate its first degree Hochschild cohomology group. We further determine the group of algebra automorphisms for the two-parameter Hopf algebra. As a result, we determine the group of Hopf algebra automorphisms for. [ABSTRACT FROM PUBLISHER]

We obtain certain results on a finitep-group whose central automorphisms are all class preserving. In particular, we prove that ifGis a finitep-group whose central automorphisms are all class preserving, thend(G) is even, whered(G) denotes the number of elements in any minimal generating set forG. As an application of these results, we obtain some results regarding finitep-groups whose automorphisms are all class preserving. [ABSTRACT FROM PUBLISHER]

We introduce a notion of natural orderings of elements of connected finite quandles. LetQbe such a quandle of ordern. Any automophism onQis a natural ordering when the elements are already ordered naturally. Suppose that the permutation *qis a cycle of lengthn − 1. Then, the operation tables for such orderings coincide, which leads to the classification of automorphisms onQ. Moreover, every row and column of the operation table contains all the elements of such a quandle, which is due to K. Oshiro. We also consider the general case of finite connected quandles. [ABSTRACT FROM PUBLISHER]

For any object L in the category of precrossed modules in Lie algebras PXLie, we construct the object Act(L), which we call the actor of this object. From this construction, we derive the notions of action, center, semidirect product, derivation, commutator, and abelian precrossed module in PXLie. We show that the notion of action is equivalent to the one given in semi-abelian categories, and Act(L) is the split extension classifier for L. In the case of a crossed module in Lie algebras we show how to recover its actor in the category of crossed modules from its actor in the category of precrossed modules. [ABSTRACT FROM AUTHOR]

Using a unified method, we determine the structure of automorphisms and representations of arbitrary polyadic groups. More precisely, for a polyadic group (G, f) = der θ, b (G, ·), we obtain a complete description of automorphisms and representations of (G, f) in terms of automorphisms and representations of the binary group (G, ·). [ABSTRACT FROM PUBLISHER]

Let  be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero. It is proved in this article that a bijective map ϕ on  preserves Lie products if and only if it is a composition of a Lie algebra automorphism and a bijective map extended by an automorphism of the base field. [ABSTRACT FROM AUTHOR]

This article explores L∞ structures on 3-dimensional vector spaces with both - and 2-gradings. Since -graded L∞ algebras are special cases of 2-graded algebras in the induced 2-grading, there are generally fewer -graded L∞ structures on a given space. However, degree zero automorphisms (rather than even automorphisms) determine equivalence in a -graded space. We therefore find nontrivial examples in which the map from the -graded moduli space to the 2-graded moduli space is bijective, injective but not surjective, or surjective but not injective. Additionally, we study how the codifferentials in the moduli spaces deform into other nonequivalent codifferentials. [ABSTRACT FROM AUTHOR]

We prove that a Jordan (φ, θ)-derivation of a 2-torsion free semiprime ring, with θ an automorphism, must be a (φ, θ)-derivation. This is then used to show more generally that an additive mapping of a semiprime ring to itself that behaves like a generalized (φ, θ)-derivation on nth powers is, in fact, a generalized (φ, θ)-derivation of the ring. [ABSTRACT FROM AUTHOR]

Let φ be an automorphism of a free group Fn of rank n, and let Mφ = Fn ⋊φ  be the corresponding mapping torus of φ. We study the group Out(Mφ) under certain technical conditions on φ. Moreover, in the case of rank 2, we classify the cases when this group is finite or virtually cyclic, depending on the conjugacy class of the image of φ in GL2(). As an application, we solve the isomorphism problem for the family of F2-by- groups, in terms of the two defining automorphisms. [ABSTRACT FROM AUTHOR]

Let X denote the restricted Lie superalgebras of Cartan type W, S, H, or K over a field of characteristic p > 3, and 𝔄 the corresponding underlying superalgebra of X. Employing the invariance of the filtration of X we construct an isomorphism of Aut X to Aut(𝔄:X), the admissible automorphism group of the associative super-commutative superalgebra 𝔄. Moreover, it is proved that the group isomorphism above maps the standard normal series of Aut X to the one of Aut(𝔄:X), and also maps the homogeneous automorphism group of X to the admissible homogeneous automorphism group of 𝔄. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with center Z , L a noncentral Lie ideal of R , and s a nontrivial automorphism of R such that [ u s , u ] n ? Z for all u ? L . If either char( R ) > n or char( R ) = 0, then R satisfies s 4 , the standard identity in 4 variables. [ABSTRACT FROM AUTHOR]

Let F m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m , with m ≥ 2. We show that F m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F (N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F (N). We prove that the fixed point subalgebra F (N) G is not finitely generated. [ABSTRACT FROM AUTHOR]

Michio Suzuki constructed a sequence of five simple groups G i , with i = 0,..., 4, and five graphs Δ i , with i = 0,..., 4, such that Δ i appears as a subgraph of Δ i +1 for i = 0,..., 3 and G i is an automorphism group of Δ i for i = 0,..., 4. The largest group G 4 was a new sporadic group of order 448 345 497 600. It is now called the Suzuki group Suz. These groups and graphs form what Jacques Tits calls the Suzuki tower. In a previous work, we constructed a rank four geometry Γ( HJ ) on which the Hall-Janko sporadic simple group acts flag-transitively and residually weakly primitively. In this article, we show that Γ( HJ ) belongs to a family of five geometries in bijection with the Suzuki tower. The largest of them is a geometry of rank six, on which the Suzuki sporadic group acts flag-transitively and residually weakly primitively. [ABSTRACT FROM AUTHOR]

We study the conjugacy problem in cyclic extensions of free groups. It is shown that the conjugacy problem is solvable in split extensions of finitely generated free groups by virtually inner automorphisms. An algorithm for construction of the unique representative (the conjugacy normal form) for each conjugacy class is given. [ABSTRACT FROM AUTHOR]

Letω(G) denote the number of orbits on the finite group G under the action of Aut(G). Using the classification of finite simple groups, we prove that for any positive integer n, there is only a finite number of (non-abelian) finite simple groups G satisfyingω(G) ≤ n. Then we classify all finite simple groups G such thatω(G) ≤ 17. The latter result was obtained by computational means, using the computer algebra system GAP. [ABSTRACT FROM AUTHOR]

We construct an example of an IA-automorphism of the free metabelian group of rank n ≥ 3 without nontrivial fixed points. That gives a negative answer to the question raised by Shpilrain in [Shpilrain, V. (1998). Fixed points of endomorphisms of a free metabelian group. Math. Proc. Cambridge Philos. Soc. 123(1): 75-83. MR 98k:20056]. By a result of Bachmuth [Bachmuth, S. (1965). Auto-morphisms of free metabelian groups. Trans. Am. Math. Soc. 118:93-1104. MR 31 #483 1], such an automorphism does not exist if the rank is equal to 2. [ABSTRACT FROM AUTHOR]

We study the classification of those finite groups G having a non-inner class preserving automorphism. Criteria for these automorphisms to be inner are established. Let G be a nilpotent-by-nilpotent group and S ∈ Syl2(G). If S is abelian, generalized quaternion or S is dihedral, and in this case G is also metabelian, then Outc(G) = 1. If S is generalized quaternion F(S) ⊂ F(G) and S4 is not a homomorphic image of G, then Outc(G) = 1. As a consequence, it follows that the normalizer problem of group rings has a positive answer for these groups. [ABSTRACT FROM AUTHOR]

Let R be a ring with identity, and let U be a subgroup of the unit group of R. Denote by R+ the underlying additive group of R. Consider the action of U on R+ given by left multiplication, and the corresponding semidirect product group R+ ⋊ U. We explicitly compute the automorphism group and the outer automorphism group of R+ ⋊ U, under certain assumptions on R and U. The groups of outer automorphisms of the three euclidean triangle groups, which motivated our investigations, are obtained as particular cases. Many other examples to which our method applies are given. [ABSTRACT FROM AUTHOR]

In this article, we explicitly calculate the McKay-Thompson series corresponding to 3A, 7A, 3B, 5B and 15C elements of the Monster simple group by using the framed VOA structure of the Moonshine VOA and automorphisms which permute 48 mutually orthogonal rational conformal vectors with central charge 1/2. [ABSTRACT FROM AUTHOR]

We consider the group G of C-automorphisms of C(x, y) (resp. C[x, y]) generated by s, t such that t(x) = y, t(y) = x and s(x) = x, s(y) = -y + u(x) where u &epsis; C[x] is of degree k ≥ 2. Using Galois's theory, we show that the invariant field and the invariant algebra of G are equal to C. [ABSTRACT FROM AUTHOR]

In the theory of Hopf algebras, grouplike elements are well studied. We argue that this notion is too restrictive when dealing with multiplier Hopf algebras. We introduce the “intrinsic” group, situated in the multiplier algebra of the multiplier Hopf algebra. When dealing with an algebraic quantum group A, the intrinsic group of the dual  characterizes a special class of automorphisms on A. For multiplier Hopf algebras which are paired in the sense of Drabant and Van Daele. (Drabant, B., Van Daele, A. Pairing and quantum double for multiplier Hopf algebras. (2001). Algebras and Representation Theory 4:109–132), we prove that the intrinsic group characterizes the semi-invariants of the action associated to the pairing. [ABSTRACT FROM AUTHOR]

The automorphism group of C[T]/(T[sup m])[X[sub 1], …, X[sub n]] is studied, and a sufficient set of generators is given. Motivations for this theorem are given. [ABSTRACT FROM AUTHOR]