On Closed Subsets of Free Groups (original) (raw)
Profinite Topologies in Free Products of Groups
International Journal of Algebra and Computation, 2004
Let [Formula: see text] be a nonempty class of finite groups closed under taking subgroups, quotients and extensions. We consider groups G endowed with their pro-[Formula: see text] topology, and say that G is 2-subgroup separable if whenever H and K are finitely generated closed subgroups of G, then the subset HK is closed. We prove that if the groups G1 and G2 are 2-subgroup separable, then so is their free product G1*G2. This extends a result to T. Coulbois. The proof uses actions of groups on abstract and profinite trees.
On Finitely Generated Subgroups of Free Products
Journal of the Australian Mathematical Society, 1971
If H is a subgroup of a group G we shall say that G is H-residually finite if for every element g in G, outside H, there is a subgroup of finite index in G, containing H and still avoiding g. (Then, according to the usual definition, G is residually finite if it is E-residually finite, where E is the identity subgroup). Definitions of other terms used below may be found in § 2 or in [6].
Some nonprojective subgroups of free topological groups
Proceedings of the American Mathematical Society, 1975
For the free topological group on an interval [a, b] a family of closed, locally path-connected subgroups is given such that each group is not projective and so not free topological. Simplicial methods are used, and the test for nonprojectivity is nonfreeness of the group of path components. Similar results are given for the abelian case.
Extending partial automorphisms and the profinite topology on free groups
Transactions of the American Mathematical Society, 1999
A class of structures C is said to have the extension property for partial automorphisms (EPPA) if, whenever C 1 and C 2 are structures in C, C 1 finite, C 1 ⊆ C 2 , and p 1 , p 2 ,. .. , pn are partial automorphisms of C 1 extending to automorphisms of C 2 , then there exist a finite structure C 3 in C and automorphisms α 1 , α 2 ,. .. , αn of C 3 extending the p i. We will prove that some classes of structures have the EPPA and show the equivalence of these kinds of results with problems related with the profinite topology on free groups. In particular, we will give a generalisation of the theorem, due to Ribes and Zalesskiȋ stating that a finite product of finitely generated subgroups is closed for this topology.
Characterization of Bases of Subgroups of Free Topological Groups
Journal of the London Mathematical Society, 1983
It is shown here that if Y is a closed subspace of the free topological group FM (X) on a fc^-space X then FM {X) has a closed subgroup topologically isomorphic to FM (Y). Thus the problem of determining whether FM (Y) can be embedded in FM (X) is reduced to that of checking if FM (X) contains a closed copy of y. As an extension of the above result it is shown that if A y , A 2 ,..., A n are (not necessarily distinct) closed subspaces of FM (X), where X is a fc^-space, then FM {X) has a closed subgroup topologically isomorphic to FM (^ x A 2 x ... x A n).
A remark on free topological groups with no small subgroups
Journal of the Australian Mathematical Society, 1974
For a completely regular space X let G(X) be the Graev free topological group on X. While proving G(X) exists for completely regular spaces X, Graev showed that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). The free group topology on G(X) is usually strictly finer than this pseudo-metric topology. In particular this is the case when X is not totally disconnected (see Morris and Thompson [7]). It is of interest to know when G(X) has no small subgroups (see Morris [5]). Morris and Thompson [6] showed that this is the case if and only if X admits a continuous metric. The proof relied on properties of the free group topology and it is natural to ask if G(X) with its pseudo-metric topology has no small subgroups when and only when X admits a continuous metric. We show that this is the case. Topological properties of G(X) associated with the pseudo-metric topology have recently been studied by Joiner [3] and Abels [1].
On the free profinite products of profinite groups with commuting subgroups
International Journal of Group Theory, 2016
In this paper we introduce the construction of free profinite products of profinite groups with commuting subgroups. We study a particular case: the proper free profinite products of profinite groups with commuting subgroups. We prove some conditions for a free profinite product of profinite groups with commuting subgroups to be proper. We derive some consequences. We also compute profinite completions of free products of (abstract) groups with commuting subgroups.
Metrizability of subgroups of free topological groups
Bulletin of the Australian Mathematical Society, 1986
It is shown that any sequential subgroup of a free topological group is either sequential of order ω1 or discrete. Hence any metrizable subgroup of a free topological group is discrete.
The character of free topological groups II
Applied General Topology, 2005
A systematic analysis is made of the character of the free and free abelian topological groups on metrizable spaces and compact spaces, and on certain other closely related spaces. In the first case, it is shown that the characters of the free and the free abelian topological groups on X are both equal to the "small cardinal" d if X is compact and metrizable, but also, more generally, if X is a non-discrete k!-space all of whose compact subsets are metrizable, or if X is a non-discrete Polish space. An example is given of a zero-dimensional separable metric space for which both characters are equal to the cardinal of the continuum. In the case of a compact space X, an explicit formula is derived for the character of the free topological group on X involving no cardinal invariant of X other than its weight; in particular the character is fully determined by the weight in the compact case. This paper is a sequel to a paper by the same authors in which the characters of the free groups were analysed under less restrictive topological assumptions.
Inverse automata and profinite topologies on a free group
Journal of Pure and Applied Algebra, 2002
This paper gives an elementary, self-contained proof that a finite product of finitely generated subgroups of a free group is closed in the profinite topology. The proof uses inverse automata (graph immersions) and inverse monoid theory. Generalizations are given to other topologies. In particular, we obtain the new result that, for arborescent pseudovarieties, the product of two closed finitely generated subgroups is again closed. An application to monoid theory is given.
On groups whose subgroups are closed in the profinite topology
Journal of Pure and Applied Algebra, 2009
A group is called extended residually finite (ERF) if every subgroup is closed in the profinite topology. The ERF-property is studied for nilpotent groups, soluble groups, locally finite groups and FC-groups. A complete characterization is given of FC-groups which are ERF.
Free products of topological groups
Bull. Austral. Math. Soc., 1971
In this note the notion of a free topological product | | G of a set {G } of topological groups is introduced. It is shown that it always exists, is unique and is algebraically isomorphic to the usual free product of the underlying groups. Further if each G is Hausdorff, then ] | G is Hausdorff and each G is a closed subgroup. Also ] \ G is a free topological group (respectively, maximally almost periodic) if each G is. This notion is then combined with the theory of varieties of topological groups developed by the author. For Y_ a variety of topological groups, the ^-product of groups in J/ is defined. It is shown that the V-product, V, 1 I ^ °^ a n v-ot set {G } of groups in V_ exists, is unique and is algebraically isomorphic to the usual varietal product. It is noted that the .V-product of Hausdorff groups is not necessarily Hausdorff, but is if V_ is abelian. Each G is a quotient group of V "| f G. It is proved that the V-product of free topological groups of V. and projective topological groups of .V are of the same type. Finally it is shown that V *] f G is connected if and only if each G is connected. J a
Proceedings of the American Mathematical Society, 2007
Recently, it has been shown by Harbater and Stevenson that a profinite group G G is free profinite of infinite rank m m if and only if G G is projective and m m -quasifree. The latter condition requires the existence of m m distinct solutions to certain embedding problems for G G . In this paper we provide several new non-trivial examples of m m -quasifree groups, projective and non-projective. Our main result is that open subgroups of m m -quasifree groups are m m -quasifree.
A Study of finitely generated Free Groups via the Fundamental Groups
arXiv: Algebraic Topology, 2017
Free groups have many applications in Algebraic Topology. In this paper I specifically study the finitely generated free groups by using the covering spaces and fundamental groups. By the Van Kampen's theorem, we have a famous fact that the fundamental group of a wedge sum of circles is a free group. Therefore, to study free groups, we could try to figure out the covering spaces of the wedge sum of circles. And in the appendix B, I prove the Nielsen-Schreier theorem which I will use this to study finitely index subgroups of a finitely generated free group.
Some Applications of Free Group
Journal of Multidisciplinary Modeling and Optimization, 2020
In this paper, we study many concepts as applications of free group, for example, presentation, rank of free group, and inverse of free group. We discussed some results about presentation concept and related it with free group. The our main result about free rank, is if G is a group, then G is free rank n if and only if G≅Zn. Also we obtained a new fact about inverse semigroup which say there is no free inverse semigroup is finitely generated as a semigroup. Moreover, we studied some results of inverse of free semigroup, These were illustrated by formulating Theorems, Lemma, Corollaries, and all of these concepts were explained through detailed examples.
The free topological group over the rationals
General Topology and its Applications, 1979
In this paper we investigate the topological structure of the Graev free topoiogical group over ;he rationals. We show that this free group fails to be a k-space and fails to carry the weak topology generalied by its subspaces of words of length less than or equal to n. As tools in this investigation we establish some properties of net convergence in free groups and also some propertics of certain canonical maps which are closely related to the topological structure of free groups. j A,M.S. (M(X) Subj. Class.: 22A99,20E05, 55D50,55620.