Subgroups Of Infinite Rank Research Articles

Groups whose proper subgroups of infinite rank have a permutability transitive relation

Abstract Let 𝐺 be a group. A subgroup 𝐻 of 𝐺 is called permutable if H ⁢ X = X ⁢ H HX=XH for all subgroups 𝑋 of 𝐺. Permutability is not in general a transitive relation, and 𝐺 is called a PT \mathrm{PT} -group if, whenever 𝐾 is a permutable subgroup of 𝐺 and 𝐻 is a permutable subgroup of 𝐾, we always have that 𝐻 is permutable in 𝐺. The property PT \mathrm{PT} is not inherited by subgroups, and 𝐺 is called a PT ̄ \overline{\mathrm{PT}} -group if all its subgroups have the PT \mathrm{PT} -property. We prove that if 𝐺 is a soluble group of infinite rank whose proper subgroups of infinite rank have the PT \mathrm{PT} -property, then 𝐺 is a PT ̄ \overline{\mathrm{PT}} -group.

Knot reversal and rational concordance

AbstractWe give an infinite family of knots that are not rationally concordant to their reverses. More precisely, if denotes the involution of the rational knot concordance group induced by reversal and denotes the subgroup of knots fixed under in , then contains an infinite rank subgroup. As a corollary, we show that there exists a knot such that for every pair of coprime integers and , the ‐cable of is not concordant to the reverse of the ‐cable of .

On groups in which subnormal subgroups of infinite rank are commensurable‎ ‎with some normal subgroup

We study soluble groups G in which each subnormal subgroup H with infinite rank is commensurable with a normal subgroup, i.e. there exists a normal subgroup N such that the intersection of H and N has finite index in both H and N. We show that if such a G is periodic, then all subnormal subgroups are commensurable with a normal subgroup, provided either the Hirsch-Plotkin radical of G has infinite rank or G is nilpotent-by-abelian (and has infinite rank).

On the Top Homology Group of the Johnson Kernel

The action of the mapping class group $\mathrm{Mod}_g$ of an oriented surface $\Sigma_g$ on the lower central series of $\pi_1(\Sigma_g)$ defines the descending filtration in $\mathrm{Mod}_g$ called the Johnson filtration. The first two terms of it are the Torelli group $\mathcal{I}_g$ and the Johnson kernel $\mathcal{K}_g$. By a fundamental result of Johnson (1985), $\mathcal{K}_g$ is the subgroup of $\mathrm{Mod}_g$ generated by all Dehn twists about separating curves. In 2007, Bestvina, Bux, and Margalit showed the group $\mathcal{K}_g$ has cohomological dimension $2g-3$. We prove that the top homology group $H_{2g-3}(\mathcal{K}_g)$ is not finitely generated. In fact, we show that it contains a free abelian subgroup of infinite rank, hence, the vector space $H_{2g-3}(\mathcal{K}_g,\mathbb{Q})$ is infinite-dimensional. Moreover, we prove that $H_{2g-3}(\mathcal{K}_g,\mathbb{Q})$ is not finitely generated as a module over the group ring $\mathbb{Q}[\mathcal{I}_g]$.

Groups whose proper subgroups of infinite rank are Černikov-by-hypercentral or hypercentral-by-Černikov

Groups whose proper subgroups of infinite rank are Černikov-by-hypercentral or hypercentral-by-Černikov

Groups with some restrictions on non-Baer subgroups

It is proved that if G is an $$\mathfrak {X}$$ -group of infinite rank whose proper subgroups of infinite rank are Baer groups, then so are all proper subgroups of G, where $$\mathfrak {X}$$ is the class defined by N.S. Cernikov as the closure of the class of periodic locally graded groups by the closure operations $$\varvec{\acute{P}}$$ , $$\varvec{\grave{P}}$$ and $$ \varvec{L}$$ . We prove also that if a locally graded group, which is neither Baer nor Cernikov, satisfies the minimal condition on non-Baer subgroups, then it is a Baer-by-Cernikov group which is a direct product of a p-subgroup containing a minimal non-Baer subgroup of infinite rank, by a Cernikov nilpotent $$p^{\prime }$$ -subgroup, for some prime p. Our last result states that a group is locally graded and has only finitely many conjugacy classes of non-Baer subgroups if, and only if, it is Baer-by-finite and has only finitely many non-Baer subgroups.

Groups with many modular or self-normalizing subgroups

In this paper, locally graded group satisfying the minimal condition on subgroups which are neither modular nor self-normalizing are described; locally (soluble-by-finite) groups of infinite rank in which every subgroup is either modular or self-normalizing are also characterized in terms of their subgroups of infinite rank. Moreover, the results obtained are used to study groups satisfying similar restrictions on subgroups which are neither permutable nor self-normalizing.

Satellites of infinite rank in the smooth concordance group

We conjecture that satellite operations are either constant or have infinite rank in the concordance group. We reduce this to the difficult case of winding number zero satellites, and use SO(3) gauge theory to provide a general criterion sufficient for the image of a satellite operation to generate an infinite rank subgroup of the smooth concordance group \({\mathcal {C}}\). Our criterion applies widely; notably to many unknotted patterns for which the corresponding operators on the topological concordance group are zero. We raise some questions and conjectures regarding satellite operators and their interaction with concordance.

Amenable signatures, algebraic solutions and filtrations of the knot concordance group

It is known that each of the successive quotient groups of the grope and solvable filtrations of the knot concordance group has an infinite rank subgroup. The generating knots of these subgroups are constructed using iterated doubling operators. In this paper, for each of the successive quotients of the filtrations we give a new infinite rank subgroup which trivially intersects the previously known infinite rank subgroups. Instead of iterated doubling operators, the generating knots of these new subgroups are constructed using the notion of algebraic $n$-solutions, which was introduced by Cochran and Teichner. Moreover, for each slice knot $K$ whose Alexander polynomial has degree greater than 2, we construct the generating knots such that they have the same derived quotients and higher-order Alexander invariants up to a certain order. In the proof, we use an $L^2$-theoretic obstruction for a knot to being $n.5$-solvable given by Cha, which is based on $L^2$-theoretic techniques developed by Cha and Orr. We also generalize the notion of algebraic $n$-solutions to the notion of $R$-algebraic $n$-solutions where $R$ is either rationals or the field of $p$ elements for a prime $p$.

Groups with an Engel restriction on proper subgroups of infinite rank

We prove that a locally graded group whose proper subgroups are Engel (respectively, [Formula: see text]-Engel) is either Engel (respectively, [Formula: see text]-Engel) or finite. We also prove that a group of infinite rank whose proper subgroups of infinite rank are Engel (respectively, [Formula: see text]-Engel) is itself Engel (respectively, [Formula: see text]-Engel), provided that [Formula: see text] belongs to the Černikov class [Formula: see text], which is the closure of the class of periodic locally graded groups by the closure operations Ṕ, P̀, R and L.

On the Upsilon invariant and satellite knots

We study the effect of satellite operations on the Upsilon invariant of Ozsvath-Stipsicz-Szabo. We obtain results concerning when a knot and its satellites are independent; for example, we show that the set $\{D_{2^i,1}\}_{i=1}^\infty$ is a basis for an infinite rank summand of the group of smooth concordance classes of topologically slice knots, for D the positive clasped untwisted Whitehead double of any knot with positive tau-invariant, e.g. the right-handed trefoil. We also prove that the image of the Mazur satellite operator on the smooth knot concordance group contains an infinite rank subgroup of topologically slice knots.

Groups with permutability conditions for subgroups of infinite rank

In this paper, the structure of non-periodic generalized radical groups of infinite rank whose subgroups of infinite rank satisfy a suitable permutability condition is investigated.

Groups in which every subgroup of infinite rank is almost permutable

ABSTRACTIn this paper, the structure of locally finite groups of infinite rank whose subgroups of infinite rank are permutable in a subgroup of finite index is investigated.

Independence of satellites of torus knots in the smooth concordance group

The main goal of this article is to obtain a condition under which an infinite collection $\mathscr{F}$ of satellite knots (with companion a positive torus knot and pattern similar to the Whitehead link) freely generates a subgroup of infinite rank in the smooth concordance group. This goal is attained by examining both the instanton moduli space over a 4-manifold with tubular ends and the corresponding Chern-Simons invariant of the adequate 3-dimensional portion of the 4-manifold. More specifically, the result is derived from Furuta's criterion for the independence of Seifert fibred homology spheres in the homology cobordism group of oriented homology 3-spheres. Indeed, we first associate to $\mathscr{F}$ the corresponding collection of 2-fold covers of the 3-sphere branched over the elements of $\mathscr{F}$ and then introduce definite cobordisms from the aforementioned covers of the satellites to a number of Seifert fibered homology spheres. This allows us to apply Furuta's criterion and thus obtain a condition that guarantees the independence of the family $\mathscr{F}$ in the smooth concordance group.

Groups with many subgroups having polycyclic-by-finite layers or derived subgroup

ABSTRACTA group is called (PF)L if the subgroups generated by its elements having same order (finite or infinite) are polycyclic-by-finite. In the present paper we prove that a group is locally graded minimal non-((PF)L∪(𝔓𝔉)𝔄) if, and only if, it is non-perfect minimal non-FC, where (𝔓𝔉)𝔄 denotes the class of (polycyclic-by-finite)-by-abelian groups. We prove also that a group of infinite rank whose proper subgroups of infinite rank are in ((PF)L∪(𝔓𝔉)𝔄) is itself in ((PF)L∪(𝔓𝔉)𝔄) provided that it is locally (soluble-by-finite) without simple homomorphic images of infinite rank. Our last result concerns groups that satisfy the minimal condition on non-((PF)L∪(𝔓𝔉)𝔄)-subgroups.

Groups in which Every Subgroup of Infinite Rank is Nearly Permutable

Groups in which Every Subgroup of Infinite Rank is Nearly Permutable

On locally finite groups whose subgroups of infinite rank have some permutable property

In this paper, we study the behavior of locally finite groups of infinite rank whose proper subgroups of infinite rank have one of the three following properties, which are generalizations of permutability: S-permutability, semipermutability and S-semipermutability. In particular, it is proved that if G is a locally finite group of infinite rank whose proper subgroups of infinite rank are S-permutable (resp. semipermutable), then G is locally nilpotent (resp. all subgroups are semipermutable). For locally finite groups whose proper subgroups of infinite rank are S-semipermutable, the same statement can be proved only for groups with min-p for every prime p. A counterexample is given for the general case.

Groups whose proper subgroups of infinite rank have minimax conjugacy classes

If [Formula: see text] is a class of groups, then a group [Formula: see text] is said to be a [Formula: see text]-group, if [Formula: see text] is a [Formula: see text]-group for all [Formula: see text]. This is a generalization of the familiar property of being an [Formula: see text]-group. In the present paper we consider a class [Formula: see text] of soluble-by-finite minimax groups such that [Formula: see text] is a subgroup closed class and if [Formula: see text] is a non-[Formula: see text]-group whose proper subgroups of infinite rank are [Formula: see text]-groups, then there exists a prime [Formula: see text] such that every finite homomorphic image of [Formula: see text] is a cyclic [Formula: see text]-group. Our main result states that if [Formula: see text] is a locally (soluble-by-finite) group of infinite rank which has no simple factor group of infinite rank and if all proper subgroups of [Formula: see text] of infinite rank are [Formula: see text]-groups, then so are all proper subgroups of [Formula: see text]. One can take for [Formula: see text] the class of finite, polycyclic-by-finite, Chernikov, reduced minimax or soluble-by-finite minimax groups.

Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes

A group $G$ is said to be a $(PF)C$-group or to have polycyclic-by-finite conjugacy classes, if $G/C_{G}(x^{G})$ is a polycyclic-by-finite group for all $xin G$. This is a generalization of the familiar property of being an $FC$-group. De Falco et al. (respectively, de Giovanni and Trombetti) studied groups whose proper subgroups of infinite rank have finite (respectively, polycyclic) conjugacy classes. Here we consider groups whose proper subgroups of infinite rank are $(PF)C$-groups and we prove that if $G$ is a group of infinite rank having a non-trivial finite or abelian factor group and if all proper subgroups of $G$ of infinite rank are $(PF)C$-groups, then so is $G$. We prove also that if $G$ is a locally soluble-by-finite group of infinite rank which has no simple homomorphic images of infinite rank and whose proper subgroups of infinite rank are $(PF)C$-groups, then so are all proper subgroups of $G$.

GROUPS OF INFINITE RANK WITH NORMALITY CONDITIONS ON SUBGROUPS WITH SMALL NORMAL CLOSURE

Groups of infinite rank in which every subgroup is either normal or contranormal are characterised in terms of their subgroups of infinite rank.