Graphs with Imprimitive Automorphism Group

We study the conservativeness property of affine algebraic groups over an algebraically closed field of characteristic 0 and of their group of automorphisms. We obtain a certain decomposition of affine algebraic groups, and this, together with the result of Hochschild and Mostow, becomes a major tool in our study of the conservativeness property of the group of automorphisms.

A power automorphism of a group G is an automorphism fixing every subgroup of G. Power automorphisms have been studied by many authors, mainly by C.D.H. Cooper [2]. The set PAutG of all power automorphisms of a group G is a normal, abelian, residually finite subgroup of the full automorphism group AutG of G. The aim of this paper is the study of quasi-power automorphisms of infinite groups. We say that an automorphism of a group G is a quasi-power automorphism if it fixes all but finitely many subgroups of G. It is clear that the set of all quasi-power automorphisms of G is a normal subgroup QAutG of AutG containing PAutG and that QAutG = AutG if G is finite. It is easily verified that quasi-power automorphisms fix all infinite subgroups (see Lemma 2.2 below). Automorphisms fixing infinite subgroups of groups have been studied by M. Curzio, S. Franciosi and F. de Giovanni [3] under the name of I-automorphisms. They prove that, under certain solubility or finiteness conditions for the group G, the group IAutG of all I-automorphisms of G coincides with PAutG, provided G is not a Cernikov group. They also give some sufficient conditions on a non-Cernikov group G to ensure the commutativity of IAutG and exhibit, by contrast, an infinite Cernikov group G such that IAutG is not abelian. Stronger results hold for quasi-power automorphisms. Indeed, if G is an infinite group, then QAutG is always abelian and residually finite, as happens for PAutG. Furthermore, it turns out that the existence of quasi-power automorphisms which are not power automorphisms affects the structure of an infinite group strongly, even if no further condition on this group is imposed. Our main result illustrating this is the following Theorem A, which also gives information on the subgroups which are not fixed under the action of quasi-power automorphisms.

An automorphism of a (profinite) group is called normal if each (closed) normal subgroup is left invariant by it. An automorphism of an abstract group is p-normal if each normal subgroup of p-power, where p is prime, is left invariant. Obviously, the inner automorphism of a group will be normal and p-normal. For some groups, the converse was stated to be likewise true. N. Romanovskii and V. Boluts, for instance, established that for free solvable pro-p-groups of derived length 2, there exist normal automorphisms that are not inner. Let N2 be the variety of nilpotent groups of class 2 and A the variety of Abelian groups. We prove the following results: (1) If p is a prime number distinct from 2, then the normal automorphism of a free pro-p-group of rank ≥2 in N2A is inner (Theorem 1); (2) If p is a prime number distinct from 2, then the p-normal automorphism of an abstract free N2A-group of rank ≥2 is inner (Theorem 2).

On automorphism groups of Toeplitz subshifts, Discrete Analysis 2017:11, 19 pp. A discrete dynamical system is a space $X$ with some kind of structure, together with a map $\sigma\colon X\to X$ that preserves the structure. (For instance, if $X$ is a topological space, then one asks for $\sigma$ to be continuous, and if $X$ is a differentiable manifold, then one asks for it to be a diffeomorphism.) Given such a system, one studies the structure of the orbits $x, \sigma x, \sigma^2x, \dots$ that are obtained by iterating the map $\sigma$. A particularly interesting subfield of dynamics is _symbolic dynamics_, where $X$ is a space of bi-infinite sequences over a finite alphabet $A$, $X$ is closed under the left shift, and $\sigma$ is that left shift. One also asks for $X$ to be closed in the topological sense: we take the discrete topology on $A$ and the product topology on $A^{\mathbb Z}$, of which $X$ is a subset. A system $(X,\sigma)$ is called a _shift space_. Such spaces can encode interesting combinatorial information, and that has led to a very fruitful interplay between combinatorics and dynamical systems. An _automorphism_ of the system $(X, \sigma)$ is a homeomorphism $\phi\colon X\to X$ that commutes with $\sigma$, and ${\rm Aut}(X, \sigma)$ denotes the group (under composition) of automorphisms of the system. The _complexity_ of a shift system ${\rm Aut}(X, \sigma)$ is the map $p\colon\mathbb N\to \mathbb N$ that counts the number of blocks of length $n$ appearing in the sequences $x\in X$. If the complexity is linear, then the automorphism group is understood for any shift ${\rm Aut}(X, \sigma)$, but beyond linear, the problem becomes complicated. For example, under mild assumptions on the shift ${\rm Aut}(X, \sigma)$, the automorphism group is not finitely generated and it contains isomorphic copies of all finite groups, countably many copies of $\mathbb Z$, and the free group on any finite number of generators. Thus while ${\rm Aut}(X, \sigma)$ is always countable, in general it can be quite complicated and difficult to compute. However, for several reasons it is desirable to do so: for example, it gives a useful invariant. This paper continues recent work on automorphism groups for various classes of shift spaces, computing the automorphism group for the class of Toeplitz shifts, a large class of shift systems frequently used to provide counterexamples in symbolic dynamics. A sequence $x\in A^{\mathbb Z}$ is _Toeplitz_ if every finite block in $x$ appears periodically, and a shift space $(X, \sigma)$ is a _Toeplitz shift_ if $X$ is the orbit closure of some Toeplitz sequence. (It is not hard to construct Toeplitz sequences that are not periodic. For one example, take $x_n$ to be the parity of $k$, where $k$ is maximal such that $2^k|n$.) This rigid structure on $X$ implies that ${\rm Aut}(X, \sigma)$ is Abelian, and this is the starting point for the classification of the automorphism groups of Toeplitz shifts. The authors start with Toeplitz shifts of subquadratic complexity, showing that the automorphism group is spanned by the roots of the shift map $\sigma$ modulo the torsion subgroup $T$ of ${\rm Aut}(X, \sigma)$. More generally, they show that if ${\rm Aut}(X, \sigma)/\langle\sigma\rangle$ is a periodic group, then the automorphism group is spanned by $T$ and the roots of the shift $\sigma$ (that is, the automorphisms $\phi$ such that $\phi^n=\sigma$ for some $n$). Under the further assumption that $T$ is trivial, they show that the automorphism group is either infinite cyclic or is not finitely generated. This method leads to examples of Toeplitz shifts whose complexity is arbitrarily close to linear, in the sense that for every $\varepsilon > 0$ the complexity satisfies the inequality $p(n)\leq Cn^{1+\varepsilon}$ for some constant $C=C_\varepsilon > 0$, such that the automorphism group is not finitely generated. Note that this result cannot be extended to linear complexity, where it is known that the automorphism group is always finitely generated. In the opposite regime, that of high complexity, the authors show that the automorphism group need not be large. Given any infinite and finitely generated Abelian group $G$ with cyclic torsion, they construct a Toeplitz shift with positive entropy (meaning that the complexity function grows exponentially) whose automorphism group is exactly $G$.

By a graph X we mean a finite set V(X), called the vertices of X, together with a set E(X), called the edges of X, consisting of unordered pairs of distinct elements of V(X). We shall indicate the unordered pairs by brackets. Two graphs X and Y are said to be isomorphic, denoted by X Y, if there is a one-to-one map a of V(X) onto V(Y) such that [aa,cab]eE(Y) if and only if [a, b] E(X). An isomorphism of X onto itself is said to be an automorphism of X. For each given graph X there is a group of automorphisms, denoted by G(X), where the multiplication is the multiplication of permutations. The complementary graph Xc of X is the graph whose V(XC) = V(X), and E(XC) consists of all possible edges which do not belong to E(X). It is easy to see that X and Xc have the same group of automorphisms. A graph consisting of isolated vertices only is called the null graph, and its complementary graph is called the complete graph. Both the null graph and the complete graph of n vertices have Sn, the symmetric group of n letters, as their group of automorphisms. A regular graph of degree k is a graph such that the number of edges incident with each vertex is k. The null graphs and the complete graphs are regular. The graph X is necessarily regular if G(X) is transitive. In [7], K6nig proposed the following question: When can a given abstract group be set up as the group of automorphisms of a graph? The question can be interpreted in two ways. (a) Given a finite group G, can one construct a graph whose group of automorphisms is abstractly isomorphic to G? (b) Given a permutation group G acting on n letters, can one construct a graph of n vertices whose group of automorphisms is G? The former has been answered affirmatively by Frucht in [4] and [5], and many others. Concerning the latter, Kagno in [6] investigated the graphs of vertices < 6 and their group of automorphisms. It is known that not every group can have a graph in the sense of (b). For instance, letting

We prove that the group of tame automorphisms of the universal enveloping algebra of the three-dimensional nilpotent nonabelian Lie algebra over a field of characteristic 0 is dense with respect to the formal power series topology in the group of all automorphisms. *Partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research. E-mail: drensky@math.bas.bg

In quantum logics, the notions of strong and full order determination and unitality for states on orthomodular posets are well known. These notions are defined for hypergraphs and their state spaces in a consistent manner and the relations between them and to the notions defined for orthomodular posets are discussed. The state space of a hypergraph is a polytope. This polytope is a simplex if and only if every superposition of pure states is a mixture of these same pure states. Isomorphic hypergraphs have convexly isomorphic state spaces. A class of hypergraphs is given whose group of automorphisms is group-isomorphic to the group of convex automorphisms of their state spaces.

2. Notation and definitions. All groups considered are finite and are written multiplicatively. E denotes the trivial group and 1 is used for the identity of all groups. Other notation is standard (as found, for example, in [8]). Let A and B be nontrivial abstract groups, and let F=AB be the direct sum of copies of A indexed by the set B. Explicitly, F is the set of all functions from B into A, made into a group by componentwise multiplication. ForfEF and bCB, definef Fbyf(y) =f(yb-1) for all yEB. Then for each bEB, the mapping f--+fb is an automorphism of F, and the group of all such automorphisms is isomorphic to B. The (standard restricted) wreath product of A and B, A wr B, is defined to be the group generated by F and B with relations b'lfb =fb for all f EF, beB. Here F A wr B, and A wr B is a splitting extension of F by the group of automorphisms B. W will be used throughout to designate A wr B, F (called the base group) will always be as defined above, and K will denote the subgroup of W defined by K = {f E F:xeBf(x) EA'}. We say that a group G is supersolvable if and only if it has an invariant series whose factors are of prime order, that is, a series G =Gn Gn-1, ... * Go = E with each G1 Il G and each Gj+1/Gj of prime

The relationship between ideals I of Turing degrees and groups of I-recursive automorphisms of the ordering on rationals is studied. We discuss the differences between such groups and the group of all automorphisms, prove that the isomorphism type of such a group completely defines the ideal I, and outline a general correspondence between principal ideals of Turing degrees and the first-order properties of such groups.

Let G be the group of all homeomorphisms of the unit interval onto itself. Schreier and Ulam [1] have obtained a few results, but beyond this there is virtually nothing in the literature concerning the algebraic structure of G. In this paper we shall analyze this structure in some detail. A brief outline follows. In ?2, among some results on representation of group elements by involutions and translations (elements with no interior fixed points), we prove that every translation is a product of two involutions and that every element is a product of at most four involutions. There are also theorems about certain one-parameter subgroups and about the center and commutator of G. ?3 contains the Signature Theorem, which gives a useful criterion for the conjugacy in G of two flows (orientation-preserving elements). In ?4 we enumerate completely the normal subgroups of G and of F (the flows). ?5 contains a proof that all the automorphisms of G are inner. We then show (?6) how G can be characterized in purely group-theoretic terms. This development permits some generalization to transformation groups on arbitrary ordered spaces. Finally, in ?7, we prove that every homeomorphism of the circle is a product of at most three involutions. It should be remarked that we treat G as an abstract group, without any topology. We now give some definitions and notation. Let X be an arbitrary set, and let 11(X) denote the group of all permutations of X under the law of composition (fg)(x) = f(g(x)), x e X, f, g e 11(X). If X is a topological space, G(X) will denote the subgroup of 11(X) consisting of the homeomorphisms of X onto itself. For I, the closed unit interval, I its interior, I* the real line, and C, the unit circle, the corresponding groups are G = G(I), Go = G(I), G* = G(I*), and GC = G(C). If H is any group, Z(H) will denote its center, [H] its commutator subgroup, A (H) its group of automorphisms, and A j(H) its group of inner automorphisms. If H is a subgroup of II(X), its isotropy group at x e X is the subgroup of H, denoted by H,, consisting of all elements leaving x fixed. Forf e 11(X), K(f) is the set of all fixed points of f. In G we distinguish certain important subsets. F, the group of flows mentioned above, is of index two in G; F = Go, the isotropy group at x = 0. R, the reversals, is the other coset of F in G. For f e G, we define 5(f) = + 1 if f e F, 5(f) = -1 if f e R. Obviously a is a homomorphism. For x e I, f e F, we define the signature of f at x,

Monomial Modular Representations and Construction of the Held Group

In this note we construct a 1-complex dimensional family of (marked) Schottky groups of genus 6 with the property that every closed Riemann surface of genus 6 admitting the group A5 as conformal group of automorphisms is uniformized by one of these Schottky groups. In the algebraic limit closure of this family we describe three noded Schottky groups uniformizing the three boundary points of the pencil described by Gonzalez-Aguilera and Rodriguez. We are able to find a very particular Riemann surface of genus 6 which is a (local) extremal for a maximal set of homologically independent simple closed geodesics. We observe that it is not Wimann's curve, the only Riemann surface of genus 6 with S5 as group of conformal automorphisms. The Schottky uniformizations permit us to compute a reducible symplectic representation of A5.

Abelian groups which can occur as the automorphism groups of a finite group have been investigated by many experts. In 1975, Jonah and Konvisser constructed a group of order p to 8 whose automorphism group is Abelian of order p to 16. In 1983, MacHale showed that the Abelian p-groups which can occur as the automorphism groups of a finite group must be of order greater than p to 1. In this paper, we fix our attention on determining all Abelian p-groups of minimal order which can occur as the automorphism group of a finite group and all groups whose automorphism group is the Abelian p-group of minimal order. We obtain many results about Abelian group which can not occur as the automorphism group of a finite group. Especially, we show that for the group possessing an Abelian automorphism group if it possesses a cyclic commutator subgroup then it must be cyclic itself. By these results we conclude the following: (1) if a non-cyclic group of order p to 7 possesses an Abelian automorphism group, then its automorphism group must be of order p to 12; (2) there is no group whose automorphism group is an Abelian group of order less than or equal to p to 11, where p is not equal to 2. Using the two results and some of Morigi's results we solve some problems given by MacHale in 1983. Our techniques of proof might apply to all Abelian p-groups.

The unit sum numbers of rational groups are investigated: the importance of the prime 2 being an automorphism of the rational group is discussed and other results are achieved by considering the number and distribution of rational primes which are, or are not, automorphisms of the group. Proof is given of the existence of rational groups with unit sum numbers greater than 2 but of finite value.

It is proved that if a periodic group $$\mathfrak{G}$$ has an extremal normal divisor $$\mathfrak{N}$$ , determining a complete abelian factor group $$\mathfrak{G}/\mathfrak{N}$$ , then the center of the group $$\mathfrak{G}$$ contains a complete abelian subgroup $$\mathfrak{A}$$ , satisfying the relation $$\mathfrak{G} = \mathfrak{N}\mathfrak{A}$$ and intersecting $$\mathfrak{N}$$ on a finite subgroup. It is also established with the aid of this proposition that every periodic group of automorphisms of an extremal group $$\mathfrak{G}$$ is a finite extension of a contained in it subgroup of inner automorphisms of the group $$\mathfrak{G}$$ .