Schreier Theorem Research Articles

Second cohomology of multiplicative Lie rings and Schreier's extension theory

Our goal of this paper is to develop an analogue of the theory of group extensions for multiplicative Lie rings. We first define a factor system of pair of multiplicative Lie rings , which use to construct an extension of by . Then we state Schreier's theorem for multiplicative Lie rings. We also use this notation to introduce second cohomology group of multiplicative Lie rings. Finally, we show that the equivalence classes of multiplicative Lie ring extensions can be identified with second cohomology group , where acts on by and .

Computing finite semigroups

Using a variant of Schreier's Theorem, and the theory of Green's relations, we show how to reduce the computation of an arbitrary subsemigroup of a finite regular semigroup to that of certain associated subgroups. Examples of semigroups to which these results apply include many important classes: transformation semigroups, partial permutation semigroups and inverse semigroups, partition monoids, matrix semigroups, and subsemigroups of finite regular Rees matrix and 0-matrix semigroups over groups. For any subsemigroup of such a semigroup, it is possible to, among other things, efficiently compute its size and Green's relations, test membership, factorize elements over the generators, find the semigroup generated by the given subsemigroup and any collection of additional elements, calculate the partial order of the D-classes, test regularity, and determine the idempotents. This is achieved by representing the given subsemigroup without exhaustively enumerating its elements. It is also possible to compute the Green's classes of an element of such a subsemigroup without determining the global structure of the semigroup.

A universal Krull–Lindenbaum theorem

We formulate a natural common generalisation of Krull's theorem on prime ideals and of Lindenbaum's lemma on complete consistent theories; this has instantiations in diverse branches of algebra, such as the Artin–Schreier theorem. Following Scott we put the Krull–Lindenbaum theorem in universal rather than existential form, which move allows us to give a relatively direct proof with Raoult's Open Induction in place of Zorn's Lemma. By reduction to the corresponding theorem on irreducible ideals that is due to Noether, McCoy, Fuchs and Schmidt, we further shed light on why prime ideals occur together with transfinite methods.

Group extensions and graphs

A classical result of Gaschütz affirms that given a finite A-generated group G and a prime p, there exists a group G# and an epimorphism φ:G#⟶G whose kernel is an elementary abelian p-group which is universal among all groups satisfying this property. This Gaschütz universal extension has also been described in the mathematical literature with the help of the Cayley graph. We give an elementary and self-contained proof of the fact that this description corresponds to the Gaschütz universal extension. Our proof depends on another elementary proof of the Nielsen–Schreier theorem, which states that a subgroup of a free group is free.

Schreier's extension theory for Lie rings

The purpose of this paper is to develop an analogue to the theory of group extensions for Lie rings. Factor triple of a pair of Lie rings (L, A) is defined and used to construct extensions of A by L. This leads to Schreier's theorem for Lie rings. We also use the notion of factor triples to define the second cohomology group of (L, A) in the case where A is abelian and L acts on A.

Open subgroups of free topological groups

The theory of covering spaces is often used to prove the Nielsen–Schreier theorem, which states that every subgroup of a free group is free. We apply the more general theory of semicovering spaces to obtain analogous subgroup theorems for topologica

Determining irreducibility and ramification groups for an additive extension of the rational function field

Let K be a subfield of F ̄ p , not necessarily proper, and a( T) be an additive polynomial defined over K. Suppose that f( x)∈ K[ x] and consider the polynomial a( T)− f( x) over K( x). We provide a method to (i) determine its irreducibility and (ii) compute the ramification groups in the resulting additive extension of K( x). In several cases, our method is easier to use and provides more information than the Artin–Schreier theorem. As an application of this method, we study families of additive extensions of K( x) obtained using symmetric polynomials. We prove irreducibility and determine their ramification groups, genera, and number of rational places. We show that these families contain examples of function fields with many rational places.

Schreier theorem on groups which split over free abelian groups

Let G be either a free product with amalgamation A *c B or an HNN group A*c, where C is isomorphic to a free abelian group of finite rank. Suppose that both A and B have no nontrivial, finitely generated, normal subgroups of infinite indices. We show that if G contains a finitely generated normal subgroup N which is neither contained in C nor free, then the index of N in G is finite. Further, as an application of this result, we show that the fundamental group of a torus sum of 3-manifolds MI and M2, the interiors of which admit hyperbolic structures, have no nontrivial, finitely generated, nonfree, normal subgroup of infinite index if each of MI and M2 has at least one nontorus boundary.

Presentations for Subgroups of Monoids

The main result of this paper gives a presentation for an arbitrary subgroup of a monoid defined by a presentation. It is a modification of the well known Reidemeister–Schreier theorem for groups. Some consequences of this result are explored. It is proved that a regular monoid with finitely many left and right ideals is finitely presented if and only if all its maximal subgroups are finitely presented. An inverse monoid with finitely many left and right ideals is finitely presented as an inverse monoid if and only if it is finitely presented as a monoid. An example of a finitely presented monoid with a finitely generated but not finitely presented group of units is exhibited.

Open subgroups of free abelian topological groups

We prove that any open subgroup of the free abelian topological group on a completely regular space is a free abelian topological group. Moreover, the free topological bases of both groups have the same covering dimension. The prehistory of this result is as follows. The celebrated Nielsen–Schreier theorem states that every subgroup of a free group is free, and it is equally well known that every subgroup of a free abelian group is free abelian. The analogous result is not true for free (abelian) topological groups [1,5]. However, there exist certain sufficient conditions for a subgroup of a free topological group to be topologically free [2]; in particular, an open subgroup of a free topological group on a kω-space is topologically free. The corresponding question for free abelian topological groups asked 8 years ago by Morris [11] proved to be more difficult and remained open even within the realm of kω-spaces. In the present paper a comprehensive answer to this question is obtained.

Rings with involution as partially ordered abelian groups

Let (S, *) be a ring with involution *. The involution is positive definite if, for all finite subsets {r,-} of S, £ w * = 0 implies all the r{ are zero. Then the set of self-adjoint elements of S, denoted S, possesses a natural partial ordering, with positive cone consisting of elements of the form Z l w * ; with this ordering, S is a directed partially ordered abelian group. Let Sb denote the set of bounded elements, that is, the set of elements s such that ss* is less than an integral multiple of 1 in this ordering. Then Sb is a *-subalgebra of S whenever S is an algebra over the rationals. We will be studying the objects Sb9 S , and (Sb) , from the point of view of their ordered structures. For instance, suppose S is a field, and * is the identity. Then S is formally real, and Sb must be Prufer domain, all of whose residue fields are themselves formally real (and in fact, are embeddable in the reals). Viewing Sb as a partially ordered abelian group with order unit 1 (indeed, Sb is the convex subgroup of S generated by 1), Sb has the Riesz decomposition property, and its normalized extremal states are precisely the ring homomorphisms into the reals. There is a natural mapping from the collection of total orderings of S to the set of extremal states of Sb9 and this in turn maps to Spec Sb (the prime ideal space of Sb); when S is even a real algebra much more can be said. If either S is a field and * is not the identity, or S is a quaternionic division algebra with the natural involution, essentially the same properties hold, with the appropriate modifications. A useful tool here is an involutory version of the Artin Schreier Theorem, about the existence of sufficiently many total orderings finer than the natural ordering. Studies are made of several specific bounded subrings. For instance, if S is the rational function field in one variable over the reals, then Sb is a Dedekind domain with class group of order 2, with spectrum the unit circle (in the point-open topology), and all of its maximal ideals are not principal. Expanding the scope of S somewhat, we next allow S to be a division

Schreier theorem for analytic Moufang loops

Schreier theorem for analytic Moufang loops

A Schreier theorem for free topological groups

M.I.Graev has shown that subgroups of free topological groups need not be free. Brown and Hardy, however, have proved that any open subgroup of the free topological group on a kw-space is again a free topological group: indeed, this is true for any closed subgroup for which a Schreier transversal can be chosen continuously. This note provides a proof of this result more direct than that of Brown and Hardy. An example is also given to show that the condition stated in the theorem is not a necessary condition for freeness of a subgroup. Finally, a sharpened version of a particular case of the theorem is obtained, and is applied to the preceding example.

Conditions for a universal mapping of algebras to be a monomorphism

Introduction. We give necessary and sufficient conditions for the monicity of any given unit morphism associated with an algebraic functor and its left adjoint. The direct verification of these conditions in specific cases can be somewhat subtle. However, a stronger set of sufficient conditions for monicity may be given which is easier to check directly. The latter conditions are still sufficiently general to provide a categorical form for the proof of the Birkhoff Witt theorem, closely related to Birkhoff's original proof [3], as well as one for the Schreier theorem on free products of groups with amalgamated subgroups [1], [5].

The Schreier theorem in normal categories

The Schreier theorem in normal categories

Some theorems on the free groups of certain product varieties

It does not seem easy to improve on Theorem 1. In fact there is no complete analog of Schreier's theorem even for the free groups of product varieties. For example, suppose that G is the free metabelian group on a and b. Let N be the normal closure of a 2 and b 2. Then it is not difficult to show that G/N is the infinite dihedral group. However, as we see from Theorem 1' below, N is finitely generated. To each metabelian variety one can associate an integer, termed its

The fundamental group of a surface, and a theorem of Schreier

Schreier proved in [8] tha t a finitely generated normal subgroup U # { 1 } of a free group F is of finite index. This result was extended by Karrass and Solitar in [3], to the case when U is not necessarily normal, but contains a non-trivial normal subgroup of F. In Topology, the free groups occur as fundamental groups of surfaces with boundary, and we here extend the result still further (Theorem 6.1) to the case when F is the (non-abelian) fundamental group of any connected surface, with or without boundary, except for a Klein bottle. We use topological methods, and also the elements of Morse theory, although the latter could be ehminated. A sketch of this theory is included, however, par t ly for its intuitive appeal, and part ly because the Morse theory picks out stable generators of the fundamental group, and therefore is helpful as a tool. Indeed, the author was able to use it quickly to prove Schreier's Theorem, and Theorem 3.3 below (that an open surface has free fundamental group), before knowing tha t proofs already existed in the literature. Our exposition is always from the point of view tha t it is the surface, ra ther than the group, which is the ult imate object o f study; and we have perhaps laboured points tha t might irk the pure group-theorist.

The Reidemeister‐Schreier and Kuroš subgroup theorems

If a group G is presented in terms of generators and relations, then the classical Reidemeister-Schreier Theorem [1] gives a presentation for any subgroup of G . If G is a free product of groups G α each of which is presented in terms of generators and relations, then the main result of this paper is a presentation for any subgroup H of G , which shows the nature of H as a free product of certain subgroups of G . This result is a generalization of the celebrated Kuros Theorem [2]. It also includes the Reidemeister–Schreier Theorem and the Schreier Theorem [1] which states that any subgroup of a free group is free.