The Stabilized Automorphism Group of a Subshift

Abelian torsion-free groups of finite rank with finite automorphism groups are considered as rigid extensions of a system of strongly indecomposable groups Aj, j = 1, ..., k, of finite rank and having finite automorphism groups, by a finite p-group P. Such groups are called (A, p)-groups. The author introduces for (A, P)-groups the concept of (A, P)-type, which represents a choice of k integer matrices. A complete description of (A, P)-groups is given by means of (A, P)-types. Using this description, a series of problems on finite groups of automorphisms of torsion-free abelian groups of finite rank are solved. Furthermore, it is shown that the actual solution of any one of these problems comes down to a question of the consistency of a system of equations of the first degree modulo pt, where pt is the maximal order of elements of P. Bibliography: 11 titles.

In this chapter we develop the algebra underlying multirate filter structures over free abelian groups of finite rank. Suppose A is a free abelian group of rank N. Each subgroup Δ of A is a free abelian group of rank n ≤ N. The basis theorem describes how Δ sits in A in the sense that it characterizes the quotient group A/Δ. This quotient group can be a finite abelian group a free abelian group of finite rank the direct product of a finite abelian group and a free abelian group of finite rank.

Given a free Abelian group of finite rank, canonical subgroup matrices are defined with respect to an arbitrary but fixed basis of , which uniquely determine all subgroups of of finite index. The use of these matrices allows the classification of all subgroups of fixed finite index. An application is made in the construction of the intersection group and the union group of two arbitrary subgroups of finite index and upper and lower bounds on the index of these groups are found. Physical applications occur for rank three in the systematic symmetry analysis of domain structures, which appear in structural phase transitions in solids.

Let (X,≤) be a partially ordered set (in short, a poset). The automorphism group Aut (X,≤) is the group of all permutations g of X such that x ≤ y if and only if xg ≤ yg for all x,y∈X. We say that (X,≤) is sharply transitive if Aut(X,≤) is sharply transitive on X, that is, for x,y∈X there exists a unique g∈Aut(X,≤) with y = xg. Sharply transitive totally ordered sets have been studied by Ohkuma[4, 5], Glass, Gurevich, Holland and Shelah [3] (see also [2] and [6]). Whereas the only countable sharply transitive totally ordered set is the set of integers, there are a great variety of countable sharply transitive posets. Amongst other results, in [1] the author showed that there are countably many non-isomorphic sharply transitive posets whose automorphism group is infinite cyclic (and also gave a full description of those), whereas there are 2N0 non-isomorphic sharply transitive posets whose automorphism group is isomorphic to the additive group of the rational numbers. This suggests also that one should consider the analogous problem for free abelian groups. The purpose of this note is to show that whenever G is a countable free abelian group then there exists a sharply transitive poset whose automorphism group is isomorphic to G, and that there are already 2N0 non-isomorphic sharply transitive posets whose automorphism group is the free abelian group of rank 2.

A Borel system consists of a measurable automorphism of a standard Borel space. We consider Borel embeddings and isomorphisms between such systems modulo null sets, i.e. sets which have measure zero for every invariant probability measure. For every t>0 we show that in this category, up to isomorphism, there exists a unique free Borel system (Y,S) which is strictly t-universal in the sense that all invariant measures on Y have entropy <t, and if (X,T) is another free system obeying the same entropy condition then X embeds into Y off a null set. One gets a strictly t-universal system from mixing shifts of finite type of entropy ≥t by removing the periodic points and “restricting” to the part of the system of entropy <t. As a consequence, after removing their periodic points the systems in the following classes are completely classified by entropy up to Borel isomorphism off null sets: mixing shifts of finite type, mixing positive-recurrent countable state Markov chains, mixing sofic shifts, beta shifts, synchronized subshifts, and axiom-A diffeomorphisms. In particular any two equal-entropy systems from these classes are entropy conjugate in the sense of Buzzi, answering a question of Boyle, Buzzi and Gomez.

Let $A$ be a finite rank torsion--free abelian group. Then there exist direct decompositions $A=B\oplus C$ where $B$ is completely decomposable and $C$ has no rank 1 direct summand. In such a decomposition $B$ is unique up to isomorphism and $C$ unique up to near-isomorphism.

A BV-algebra structure on Hochschild cohomology of the integral group ring of finitely generated Abelian groups

In this article, we study 2-(v, k, λ) designs D with λ prime admitting flag-transitive and point-primitive almost simple automorphism groups G with socle T a finite exceptional simple group or a sporadic simple group. If the socle of G is a finite exceptional simple group, then we prove that D is isomorphic to one of two infinite families of 2-designs with point-primitive automorphism groups, one is the Suzuki-Tits ovoid design with parameter set (v, b, r, k, λ) = (q² + 1, q²(q² + 1)/(q − 1), q², q, q − 1), where q − 1 is a Mersenne prime, and the other is newly constructed in this paper and has parameter set (v, b, r, k, λ) = (q³(q³ − 1)/2, (q + 1)(q⁶ − 1), (q + 1)(q³ + 1), q³/2, q + 1), where q + 1 a Fermat prime. If T is a sporadic simple group, then we show that D is isomorphic to a unique design admitting a point-primitive automorphism group with parameter set (v, b, r, k, λ) = (176, 1100, 50, 2), (12, 22, 11, 6, 5) or (22, 77, 21, 6, 5).

It is proved that a suitable free Abelian group of finite rank is not absolutely closed in the class $ {\mathcal A} $ 2 of metabelian groups. A condition is specified under which a torsion-free Abelian group is not absolutely closed in $ {\mathcal A} $ 2. Also we gain insight into the question when the dominion in $ {\mathcal A} $ 2 of the additive group of rational numbers coincides with this subgroup.

The aim of this paper is to determine the structure and to establish the isomorphic invariant of the finitely generated nilpotent group G of infinite cyclic commutator subgroup. Using the structure and invariant of the group which is the central extension of a cyclic group by a free abelian group of finite rank of infinite cyclic center, we provide a decomposition of G as the product of a generalized extraspecial ℤ-group and its center. By using techniques of lifting isomorphisms of abelian groups and equivalent normal form of the generalized extraspecial ℤ-groups, we finally obtain the structure and invariants of the group G.

The power set of a group G has an induced semigroup structure, some subsets of which will form groups in their own right. We are especially interested in such subsets that are maximal. We demonstrate that even when G is a free abelian group of finite rank, the groups which arise in this way can be diverse profinite abelian groups.

In this paper, we shall discuss some recent contributions to the project [15, 14, 2, 18, 22, 23] of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite rank n ≥ 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space ℚn which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank at most n can be naturally identified with the set S (ℚn) of all nontrivial additive subgroups of ℚn. In 1937, Baer [4] solved the classification problem for the class S(ℚ)of rank 1 groups as follows.Let ℙ be the set of primes. If G is a torsion-free abelian group and 0 ≠ x ϵ G, then the p-height of x is defined to behx(p) = sup{n ϵ ℕ ∣ There exists y ϵ G such that pny = x} ϵ ℕ ∪{∞}; and the characteristic χ (x) of x is defined to be the function

It is proved that ifr1,r2, ...,rs;l1,l2, ...,lt are the ranks of the indecomposable summands of two direct decompositions of a torsion-free Abelian group of finite rank and if s0 is the number of units among the numbers ri, while t0 is the number of units among the numbers lj, thenri ⩽n - t0,lj⩽n−s0 for all i, j. Moreover, if for some i we have ri=n−t0, then among the lj's only one term is different from 1 and it is equal to n−t0; similarly if lj=n−s0 for some j. In addition, a construction is presented, allowing to form, from several indecomposable groups, a new group, called a flower group, and it is proved that a flower group is indecomposable under natural restrictions on its defining parameters.

We define a one-to-one correspondence between torsion-free Abelian groups of finite rank with marked bases and finite sequences of elements of finitely presented modules over the ring of polyadic numbers. This correspondence is a duality of two categories.

The aim of this paper is to give an upper bound for the dimension of a torus $T$ which acts on a GKM manifold $M$ effectively. In order to do that, we introduce a free abelian group of finite rank, denoted by $\mathcal{A}(\Gamma,\alpha,\nabla)$, from an (abstract) $(m,n)$-type GKM graph $(\Gamma,\alpha,\nabla)$. Here, an $(m,n)$-type GKM graph is the GKM graph induced from a $2m$-dimensional GKM manifold $M^{2m}$ with an effective $n$-dimensional torus $T^{n}$-action which preserves the almost complex structure, say $(M^{2m},T^{n})$. Then it is shown that $\mathcal{A}(\Gamma,\alpha,\nabla)$ has rank $\ell(> n)$ if and only if there exists an $(m,\ell)$-type GKM graph $(\Gamma,\widetilde{\alpha},\nabla)$ which is an extension of $(\Gamma,\alpha,\nabla)$. Using this combinatorial necessary and sufficient condition, we prove that the rank of $\mathcal{A}(\Gamma_{M},\alpha_{M},\nabla_{M})$ for the GKM graph $(\Gamma_{M},\alpha_{M},\nabla_{M})$ induced from $(M^{2m},T^{n})$ gives an upper bound for the dimension of a torus which can act on $M^{2m}$ effectively. As one of the applications of this result, we compute the rank associated to $\mathcal{A}(\Gamma,\alpha,\nabla)$ of the complex Grassmannian of 2-planes $G_{2}(\mathbb{C}^{n+2})$ with the natural effective $T^{n+1}$-action, and prove that this action on $G_{2}(\mathbb{C}^{n+2})$ is the maximal effective torus action which preserves the standard complex structure.