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The relation between cross-sectional algebras of homogeneous Banach-∗-algebraic bundles in the sense of Fell [5] and generalized L 1 -algebras, as defined in slightly different ways by Leptin [7], Busby and Smith [2], and others, has been studied by Busby in [3]. We give an extension of his result, using a different method for obtaining topological group extensions. Instead of first constructing the abstract group extension from the given factor system and then topologizing it, we work in a natural topological setting and define a topological group which turns out to be the group extension belonging to the given factor system. As a consequence we obtain (without separability assumptions) that for any measurable factor system of a locally compact group with values in some other locally compact group the corresponding abstract group extension can be topologized to give a topological (and hence locally compact) group extension.

This work relates numerical problems on matrices over the rationals to symbolic algorithms on words and finite automata. Using exact algebraic algorithms and symbolic computation, we prove new decidability results for 2 × 2 matrices over Q. Namely, we introduce a notion of flat rational sets: if M is a monoid and N ≤ M is its submonoid, then flat rational sets of M relative to N are finite unions of the form L0g1L1 ··· gtLt where all Lis are rational subsets of N and gi ∈ M. We give quite general sufficient conditions under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of GL(2, Q) over GL(2, Z) is decidable. We also show a dichotomy for nontrivial group extension of GL(2, Z) in GL(2, Q): if G is a f.g. group such that GL(2, Z) < G ≤ GL(2, Q), then either G ≅ GL(2, Z) × Zk, for some k ≥ 1, or G contains an extension of the Baumslag-Solitar group BS(1, q), with q ≥ 2, of infinite index. It turns out that in the first case the membership problem for G is decidable but the equality problem for rational subsets of G is undecidable. In the second case, decidability of the membership problem is open for every such G. In the last section we prove new decidability results for flat rational sets that contain singular matrices. In particular, we show that the membership problem is decidable for flat rational subsets of M(2, Q) relative to the submonoid that is generated by the matrices from M(2, Z) with determinants 0, ± 1 and the central rational matrices.

International audience; We investigate the relationship between (total) left- and right-orderability for monoids, in particular illustrating the finite case by various structural observations and counterexamples, also highlighting the particular role played by \emph{positive} orderability.Moreover, we construct a non-left-orderable, positively right-orderable submonoid of the free product of the cyclic group of order 7 with the free group on four generators.Any group extension of that monoid has elements of order 7.

Several number knots are defined including the five knots introduced by W. Jehne. The question of the splitting of the group extension of the Hilbert class field can be read off in terms of the triviality of these knots.

P-E. Caprace and N. Monod isolate the class $\mathscr{X}$ of locally compact groups for which relatively amenable closed subgroups are amenable. It is unknown if $\mathscr{X}$ is closed under group extension. In this note, we exhibit a large, group extension stable subclass of $\mathscr{X}$, which suggests $\mathscr{X}$ is indeed closed under group extension. Along the way, we produce generalizations of the class of elementary groups and obtain information on groups outside $\mathscr{X}$., Accepted version

The objective of this research is to analyze the effect of extension strategy and the environmental concern on knowledge about environmental issues of house wives on the slums in West Jakarta. The sample of the research were 40 housewives which selected randomly. The data were analyzed by using ANOVA with 2 x 2 factorial design. The result of the research reveals that: 1) the knowledge about environmental issues of housewives who received extension strategy by out group extension agents have not more highly than when they received extension by in group extension agents; 2) The housewives who have high environmental concern and receive extension by out group extension agents have more highly knowledge about environmental issues than when they received extension by in group exten-sion agents; 3)The housewives who have low environmental concern and received extension by out group extension agents have lower knowledge about environmen-tal issues than when they received extension by ingroup extension agents; 4) There are interaction effects between extension strategy and the environmental concern towards the housewives knowledge about environmental issues.

We relate the old and new cohomology monoids of an arbitrary monoid $M$ with coefficients in semimodules over $M$, introduced in the author's previous papers, to monoid and group extensions. More precisely, the old and new second cohomology monoids describe Schreier extensions of semimodules by monoids, and the new third cohomology monoid is related to a certain group extension problem.

We first consider two groups, F 0 = { g ∈ C [ [ z ] ] | g ( 0 ) ≠ 0 } under multiplication and F 1 = z F 0 under composition, where C [ [ z ] ] is the ring of formal power series over the complex field. It is known that the Riordan group R is isomorphic to the semidirect product F 0 ⋊ F 1 . It may be viewed as a group extension of F 1 by F 0 . In this paper, the group of three-dimensional Riordan arrays is obtained from an extension of the group R by F 0 . This concept extends to the group of multi-dimensional Riordan arrays. As an application, we illustrate the use of the three-dimensional Riordan array in multiple combinatorial sums.

We classify time-reversal breaking (class A) spinful topological crystalline insulators with crystallographic non-magnetic (32 types) and magnetic (58 types) point groups. The classification includes all possible magnetic topological crystalline insulators protected by point group symmetry. Whereas the classification of topological insulators is known to be given by the $K$-theory in the momentum space, computation of the $K$-theory has been a difficult task in the presence of complicated crystallographic symmetry. Here we consider the $K$-homology in the real space for this problem, instead of the $K$-theory in the momentum space, both of which give the same topological classification. We apply the Atiyah-Hirzebruch spectral sequence (AHSS) for computation of the $K$-homology, which is a mathematical tool for generalized (co)homology. In the real space picture, the AHSS naturally gives the classification of higher-order topological insulators at the same time. By solving the group extension problem in the AHSS on the basis of physical arguments, we completely determine possible topological phases including higher-order ones for each point group. Relationships among different higher-order topological phases are argued in terms of the AHSS in the $K$-homology. We find that in some nonmagnetic and magnetic point groups, a stack of two $\mathbb{Z}_2$ second-order topological insulators can be smoothly deformed into non-trivial fourth-order topological insulators, which implies non-trivial group extensions in the AHSS., Comment: Add E^2-pages in Appendix

We identify the class of elementary groups: the smallest class of totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contains the profinite groups and the discrete groups, is closed under group extensions of profinite groups and discrete groups, and is closed under countable increasing unions. We show this class enjoys robust permanence properties. In particular, it is closed under group extension, taking closed subgroups, taking Hausdorff quotients, and inverse limits. A characterization of elementary groups in terms of well-founded descriptive-set-theoretic trees is then presented. We conclude with three applications. We first prove structure results for general t.d.l.c.s.c. groups. In particular, we show a compactly generated t.d.l.c.s.c. group decomposes into elementary groups and topologically characteristically simple groups via group extension. We then prove two local-to-global structure theorems: Locally solvable t.d.l.c.s.c. groups are elementary and [A]-regular t.d.l.c.s.c. groups are elementary., Accepted version. To appear in The Proceedings of the London Mathematical Society

In this paper we extend and generalize the Calabi construction for flat manifolds to flat orbifolds. Let Γ be an n-dimensional crystallographic group (n-space group) that is Z -reducible. Then the flat orbifold E n / Γ has a nontrivial fibered orbifold structure. We prove that this structure can be described by a Calabi construction, that is, E n / Γ is represented as the quotient of the Cartesian product of two flat orbifolds under the diagonal action of a structure group of isometries. We determine the structure group and prove that it is finite if and only if the fibered orbifold structure has an orthogonally dual fibered orbifold structure. A fibered orbifold structure on E n / Γ corresponds to a space group extension 1 → N → Γ → Γ / N → 1 . We give a condition for the splitting of a space group extension in terms of the structure group action that is strong enough to detect the splitting of all the space group extensions corresponding to the standard Seifert fibrations of a compact, connected, flat 3-orbifold.

We define a 3-loop group $\Omega^3G$ as a subgroup of smooth maps from a 3-ball to a Lie group $G$, and then construct a 2-group based on an automorphic action on the Mickelsson-Faddeev extension of $\Omega^3G$. In this we follow the strategy of Murray et al., who earlier described a similar construction in one dimension. The three-dimensional situation presented here is further complicated by the fact that the 3-loop group extension is not central., Comment: 13 pages

It is well-known that a finite group possesses a universal central extension if and only if it is a perfect group. Similarly, given a prime number p, we show that a finite group possesses a universal p′-central extension if and only if the p′-part of its abelianization is trivial. This question arises naturally when working with group representations over a field of characteristic p.