Intervals In Subgroup Lattices Research Articles

OVERGROUPS OF WEAK SECOND MAXIMAL SUBGROUPS

A subgroup $H$ is called a weak second maximal subgroup of $G$ if $H$ is a maximal subgroup of a maximal subgroup of $G$. Let $m(G,H)$ denote the number of maximal subgroups of $G$ containing $H$. We prove that $m(G,H)-1$ divides the index of some maximal subgroup of $G$ when $H$ is a weak second maximal subgroup of $G$. This partially answers a question of Flavell [‘Overgroups of second maximal subgroups’, Arch. Math.64(4) (1995), 277–282] and extends a result of Pálfy and Pudlák [‘Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups’, Algebra Universalis11(1) (1980), 22–27].

Intervals in subgroup lattices of countable locally finite groups

We prove that every algebraic lattice with at most countably many compact elements is isomorphic to an interval in the subgroup lattice of a countable locally finite group.

On intervals in subgroup lattices of finite groups

We investigate the question of which finite lattices L L are isomorphic to the lattice [ H , G ] [H,G] of all overgroups of a subgroup H H in a finite group G G . We show that the structure of G G is highly restricted if [ H , G ] [H,G] is disconnected. We define the notion of a “signalizer lattice" in H H and show for suitable disconnected lattices L L , if [ H , G ] [H,G] is minimal subject to being isomorphic to L L or its dual, then either G G is almost simple or H H admits a signalizer lattice isomorphic to L L or its dual. We use this theory to answer a question in functional analysis raised by Watatani.

Topology of order complexes of intervals in subgroup lattices

We conjecture that the order complex of an open interval in the subgroup lattice of a finite group has the homotopy type of a wedge of spheres and prove that if ( H, G) is a minimal counterexample to this conjecture then either G is almost simple or G= HN, where N is the unique minimal normal subgroup of G, N is non-Abelian and H∩ N=1.

On Representing Finite Lattices as Intervals in Subgroup Lattices of Finite Groups

Let Mnbe the lattice of length 2 withn≥1 atoms. It is an open problem to decide whether or not every such lattice (or indeed whether or not every finite lattice) can be represented as an interval in the subgroup lattice of some finite group. We complete the work of the second author, Lucchini, by reducing this problem to a series of questions concerning the finite non-abelian simple groups.

Representation of Certain Lattices as Intervals in Subgroup Lattices

Representation of Certain Lattices as Intervals in Subgroup Lattices

Intervals in subgroup lattices of finite groups

Intervals in subgroup lattices of finite groups

Intervals in subgroup lattices of infinite groups

We prove that every algebraic lattice is isomorphic to an interval in the subgroup lattice of an infinite group. This leads to a complete lattice-theoretical characterization of intervals in subgroup lattices of infinite groups.

On Feit's examples of intervals in subgroup lattices

On Feit's examples of intervals in subgroup lattices

Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups

Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups