Intersection Of Subgroups Research Articles

Intersection configurations in free and free times free-abelian groups

In this paper, we study intersection configurations – which describe the behaviour of multiple (finite) intersections of subgroups with respect to finite generability – in the realm of free and free times free-abelian (FTFA) groups. We say that a configuration is realizable in a group $G$ if there exist subgroups $H_1,\ldots, H_k \leqslant G$ realizing it. It is well known that free groups ${\mathbb {F}_{n}}$ satisfy the Howson property: the intersection of any two finitely generated subgroups is again finitely generated. We show that the Howson property is indeed the only obstruction for multiple intersection configurations to be realizable within nonabelian free groups. On the contrary, FTFA groups ${\mathbb {F}_{n}} \times \mathbb {Z}^m$ are well known to be non-Howson. We also study multiple intersections within FTFA groups, providing an algorithm to decide, given $k\geq 2$ finitely generated subgroups, whether their intersection is again finitely generated and, in the affirmative case, compute a ‘basis’ for it. We finally prove that any intersection configuration is realizable in an FTFA group ${\mathbb {F}_{n}} \times \mathbb {Z}^m$, for $n\geq 2$ and large enough $m$. As a consequence, we exhibit finitely presented groups where every intersection configuration is realizable.

Finite Abelian groups with positive genus subgroup intersection graphs

The intersection graph of subgroups of a finite group [Formula: see text] is a graph whose vertices are all nontrivial subgroups of [Formula: see text] and in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The non-orientable genus of a graph [Formula: see text] is the smallest positive integer [Formula: see text] such that [Formula: see text] can be embedded on [Formula: see text] ([Formula: see text]), where [Formula: see text] and [Formula: see text] are the surface obtained from the sphere by attaching [Formula: see text] handles and the sphere with [Formula: see text] added crosscaps, respectively. In this paper, we classify all finite abelian groups whose non-oritentable genus of intersection graphs of subgroups are 1–3, respectively.

Subgroups of products of para τ-discrete semitopological groups

In this article we define a new class of topological spaces called para ?-discrete spaces and give an internal characterization of a subgroup of product of para ?-discrete semitopological groups having character less than or equal to ?. Also we give a partial solution of an open problem posed by S?nchez [5, Problem 3.8].

Stallings automata for free-times-abelian groups: intersections and index

We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which - as it happens in the free group - is computable in the finitely generated case. This approach provides a neat geometric description of (even non finitely generated) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric solution to the subgroup intersection problem and the finite index problem, providing recursive bases and transversals respectively.

Singular intersections of subgroups and character varieties

We prove a global local rigidity result for character varieties of 3-manifolds into \(\mathrm {SL}_2({\mathbb {C}})\). Given a 3-manifold with toric boundary M satisfying some technical hypotheses, we prove that all but a finite number of its Dehn fillings \(M_{p/q}\) are globally locally rigid in the following sense: every irreducible representation \(\rho :\pi _1(M_{p/q})\rightarrow \mathrm {SL}_2({\mathbb {C}})\) is infinitesimally rigid, meaning that \(H^1(M_{p/q},\text {Ad}_\rho )=0\). This question arose from the study of asymptotics problems in topological quantum field theory developed in Charles and Marché (Knot state asymptotics II. Irreducible representations and the Witten conjecture, 2011). The proof relies heavily on recent progress in diophantine geometry and raises new questions of Zilber–Pink type. The main step is to show that a generic curve lying in a plane multiplicative torus intersects transversally almost all subtori of codimension 1. We prove an effective result of this form, based mainly on a height upper bound of Habegger.

Some Topological and Polynomial Indices (Hosoya and Schultz) for the Intersection Graph of the Subgroup of〖 Z〗_(r^n )

Let be any group with identity element (e) . A subgroup intersection graph of a subset is the Graph with V ( ) = - e and two separate peaks c and d contiguous for c and d if and only if , Where is a Periodic subset of resulting from . We find some topological indicators in this paper and Multi-border (Hosoya and Schultz) of , where , is aprime number.

Permutation group algorithms based on directed graphs

We introduce a new framework for solving an important class of computational problems involving finite permutation groups, which includes calculating set stabilisers, intersections of subgroups, and isomorphisms of combinatorial structures. Our techniques are inspired by and generalise ‘partition backtrack’, which is the current state-of-the-art algorithm introduced by Jeffrey Leon in 1991. But, instead of ordered partitions, we use labelled directed graphs to organise our backtrack search algorithms, which allows for a richer representation of many problems while often resulting in smaller search spaces. In this article we present the theory underpinning our framework, we describe our algorithms, and we show the results of some experiments. An implementation of our algorithms is available as free software in the GraphBacktracking package for GAP.

Some Topological Indices and ( Hosoya and Schultz) Polynomial of Subgroup intersection graph of a group Z_r

Let be any group with identity element (e) . A subgroup intersection graph of a group is the Graph with V (G) = - e and two separate a vertices a and b contiguous for a and b if and only if , Where is a cyclic subgroup of generated by . We found some topological indices in this paper and (Hosoya and Schultz) Polynomial of , where , is a prime number.

Remark on subgroup intersection graph of finite abelian groups

Abstract Let G be a finite group. The subgroup intersection graph Γ ( G ) \text{Γ}(G) of G is a graph whose vertices are non-identity elements of G and two distinct vertices x and y are adjacent if and only if | 〈 x 〉 ∩ 〈 y 〉 | > 1 |\langle x\rangle \cap \langle y\rangle |\gt 1 , where 〈 x 〉 \langle x\rangle is the cyclic subgroup of G generated by x. In this paper, we show that two finite abelian groups are isomorphic if and only if their subgroup intersection graphs are isomorphic.

Lower Central Series of Baumslag–Solitar Groups

We look at Baumslag–Solitar groups. Lower central series are given for residually nilpotent Baumslag–Solitar groups. For some Baumslag–Solitar groups that are not residually nilpotent, we find the intersection of all terms of the lower central series. Also we point out non-Abelian Baumslag–Solitar groups having lower central series of length 2. For some Baumslag–Solitar groups, a connection is found between an intersection of subgroups of finite index and an intersection of terms of the lower central series.

Thickness of the subgroup intersection graph of a finite group

<abstract> <p>Let $ G $ be a finite group. The intersection graph of subgroups of $ G $ is a graph whose vertices are all non-trivial subgroups of $ G $ and in which two distinct vertices $ H $ and $ K $ are adjacent if and only if $ H\cap K\neq 1 $. In this paper, we classify all finite abelian groups whose thickness and outerthickness of subgroup intersection graphs are 1 and 2, respectively. We also investigate the thickness and outerthickness of subgroup intersection graphs for some finite non-abelian groups.</p> </abstract>

Realizable ranks of joins and intersections of subgroups in free groups

The famous Hanna Neumann Conjecture (now the Friedman–Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups [Formula: see text] and [Formula: see text] of a non-abelian free group. It is an interesting question to “quantify” this bound with respect to the rank of [Formula: see text], the subgroup generated by [Formula: see text] and [Formula: see text]. We describe a set of realizable values [Formula: see text] for arbitrary [Formula: see text], [Formula: see text], and conjecture that this locus is complete. We study the combinatorial structure of the topological pushout of the core graphs for [Formula: see text] and [Formula: see text] with the help of graphs introduced by Dicks in the context of his Amalgamated Graph Conjecture. This allows us to show that certain conditions on ranks of [Formula: see text], [Formula: see text] are not realizable, thus resolving the remaining open case [Formula: see text] of Guzman’s “Group-Theoretic Conjecture” in the affirmative. This in turn implies the validity of the corresponding “Geometric Conjecture” on hyperbolic 3-manifolds with a 6-free fundamental group. Finally, we prove the main conjecture describing the locus of realizable values for the case when [Formula: see text].

Dominating sets in intersection graphs of finite groups

Let $G$ be a group. The intersection graph $\Gamma (G)$ of $G$ is an undirected graph without loops and multiple edges, defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$ and $K$ if and only if $H\cap K \neq 1$, where $1$ denotes the trivial subgroup of $G$. In this paper, we study the dominating sets in intersection graphs of finite groups. We classify abelian groups by their domination number and find upper bounds for some specific classes of groups. Subgroup intersection is related to Burnside rings. We introduce the notion of an intersection graph of a $G$-set (somewhat generalizing the ordinary definition of an intersection graph of a group) and establish a general upper bound for the domination number of $\Gamma (G)$ in terms of subgroups satisfying a certain property in the Burnside ring. The intersection graph of $G$ is the $1$-skeleton of the simplicial complex. We name this simplicial complex intersection complex of $G$ and show that it shares the same homotopy type with the order complex of proper non-trivial subgroups of $G$. We also prove that, if the domination number of $\Gamma (G)$ is 1, then the intersection complex of $G$ is contractible.

Intersection problem for Droms RAAGs

We solve the subgroup intersection problem (SIP) for any RAAG [Formula: see text] of Droms type (i.e. with defining graph not containing induced squares or paths of length [Formula: see text]): there is an algorithm which, given finite sets of generators for two subgroups [Formula: see text], decides whether [Formula: see text] is finitely generated or not, and, in the affirmative case, it computes a set of generators for [Formula: see text]. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian groups. We note that most of RAAGs are not Howson, and many (e.g. [Formula: see text]) even have unsolvable SIP.

On joins and intersections of subgroups in free groups

We study graphs of (generalized) joins and intersections of finitely generated subgroups of a free group. We show how to disprove a lemma of Imrich and Muller on these graphs and how to repair this lemma.

New refiners for permutation group search

Partition backtrack is the current generic state of the art algorithm to search for subgroups of a given permutation group. We describe an improvement of partition backtrack for set stabilizers and intersections of subgroups by using orbital graphs. With extensive experiments we demonstrate that our methods improve performance of partition backtrack – in some cases by several orders of magnitude.

The intersection of subgroups in free groups and linear programming

We study the intersection of finitely generated subgroups of free groups by utilizing the method of linear programming. We prove that if $$H_1$$ is a finitely generated subgroup of a free group F, then the WN-coefficient $$\sigma (H_1)$$ of $$H_1$$ is rational and can be computed in deterministic exponential time in the size of $$H_1$$ . This coefficient $$\sigma (H_1)$$ is the minimal nonnegative real number such that, for every finitely generated subgroup $$H_2$$ of F, it is true that $${\bar{\mathrm {r}}}(H_1, H_2) \le \sigma (H_1) {\bar{ \mathrm {r}}}(H_1) {\bar{\mathrm {r}}}(H_2)$$ , where $${\bar{ \mathrm{r}}} (H) := \max ( \mathrm{r} (H)-1,0)$$ is the reduced rank of H, $$\mathrm {r}(H)$$ is the rank of H, and $${\bar{\mathrm {r}}}(H_1, H_2)$$ is the reduced rank of the generalized intersection of $$H_1$$ and $$H_2$$ . We also show the existence of a subgroup $$H_2^* = H_2^*(H_1)$$ of F such that $${\bar{\mathrm {r}}}(H_1, H_2^*) = \sigma (H_1) {\bar{\mathrm {r}}}(H_1) {\bar{ \mathrm {r}}}(H_2^*)$$ , the Stallings graph $$\Gamma (H_2^*)$$ of $$H_2^*$$ has at most doubly exponential size in the size of $$H_1$$ and $$\Gamma (H_2^*)$$ can be constructed in exponential time in the size of $$H_1$$ .

On a conjecture of Imrich and Müller

Abstract A conjecture of Imrich and Müller on the rank of the intersection of subgroups of free groups is disproved.

Finiteness results for subgroups of finite extensions

We discuss in the context of finite extensions two classical theorems of Takahasi and Howson on subgroups of free groups. We provide bounds for the rank of the intersection of subgroups within classes of groups such as virtually free groups, virtually nilpotent groups or fundamental groups of finite graphs of groups with virtually polycyclic vertex groups and finite edge groups. As an application of our generalization of Takahasi's Theorem, we provide a uniform bound for the rank of the periodic subgroup of any endomorphism of the fundamental group of a given finite graph of groups with finitely generated virtually nilpotent vertex groups and finite edge groups.

On the intersection of conjugate nilpotent subgroups in permutation groups

We prove that, with exception of finitely many explicitly indicated cases, for a nilpotent subgroup in a symmetric or an alternating group there always exists a subgroup conjugate to it such that the intersection of the subgroups is trivial.