The Chermak–Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups. We prove that the Chermak–Delgado lattice of a central product contains the product of ...the Chermak–Delgado lattices of the relevant central factors. Furthermore, we obtain information about heights of elements in the Chermak–Delgado lattice relative to their heights in the Chermak–Delgado lattices of central factors. We also explore how the central product can be used as a tool in investigating Chermak–Delgado lattices.
Throughout this paper, G always denotes a group and L(G) is the lattice of all subgroups of G.
If K⊴H≤G, then H/K is called a section of G; such a section is called normal if K,H⊴G. We call any set Σ ...of normal sections of G a stratification of G provided: (i) Σ is G-closed, that is, H/K∈Σ whenever H/K≃GT/L∈Σ, and (ii) L/K,H/L∈Σ for each triple K<L<H, where H/K∈Σ and L⊴G.
Now let Σ be any stratification of G. Then we write LΣ(G) to denote the set of all subgroups A of G such that AG/AG∈Σ.
We say that a stratification Σ of G is a formation Fitting set of G provided: (i) H/(K∩N)∈Σ for every two sections H/K,H/N∈Σ, and (ii) HV/K∈Σ for every two sections H/K,V/K∈Σ.
We prove that the set LΣ(G) forms a sublattice of L(G) for every formation Fitting set Σ of G and we also discuss some applications of sublattices of this kind.
Given a finite group R, we let Sub(R) denote the collection of all subgroups of R. We show that |Sub(R)|<c⋅|R|log2(|R|)4, where c<7.372 is an explicit absolute constant. This result is ...asymptotically best possible. Indeed, as |R| tends to infinity and R is an elementary abelian 2-group, the ratio|Sub(R)||R|log2(|R|)4 tends to c.
In this paper G always denotes a group. If K and H are subgroups of G, where K is a normal subgroup of H, then the factor group of H by K is called a section of G. Such a section is called normal, if ...K and H are normal subgroups of G, and trivial, if K and H are equal. We call any set S of normal sections of G a stratification of G, if S contains every trivial normal section of G, and we say that a stratification S of G is G-closed, if S contains every such a normal section of G, which is G-isomorphic to some normal section of G belonging S. Now let S be any G-closed stratification of G, and let L be the set of all subgroups A of G such that the factor group of V by W, where V is the normal closure of A in G and W is the normal core of A in G, belongs to S. In this paper we describe the conditions on S under which the set L is a sublattice of the lattice of all subgroups of G and we also discuss some applications of this sublattice in the theory of generalized finite T-groups.
Abstract The Möbius function of the subgroup lattice of a finite group has been introduced by Hall and applied to investigate several questions. In this paper, we consider the Möbius function defined ...on an order ideal related to the lattice of the subgroups of an irreducible subgroup G of the general linear group $$\textrm{GL}(n,q)$$ GL ( n , q ) acting on the n -dimensional vector space $$V=\mathbb {F}_q^n$$ V = F q n , where $$\mathbb {F}_q$$ F q is the finite field with q elements. We find a relation between this function and the Euler characteristic of two simplicial complexes $$\Delta _1$$ Δ 1 and $$\Delta _2$$ Δ 2 , the former raising from the lattice of the subspaces of V , the latter from the subgroup lattice of G .
This article provides an overview of some recent results and ideas related to the study of finite groups depending on the restrictions on some systems of their sections. In particular, we discuss ...some properties of the lattice of all subgroups of a finite group related with conditions of permutability and generalized subnormality for subgroups. The paper contains more than 30 open problems which were posed, at different times, by some mathematicians working in the discussed direction.
Let G be a finite group, μ be the Möbius function on the subgroup lattice of G, and λ be the Möbius function on the poset of conjugacy classes of subgroups of G. It was proved by Pahlings that, ...whenever G is solvable, the property
holds for any subgroup H of G. It is known that this property does not hold in general, the Mathieu group M
12
being a counterexample. In this paper we investigate the relation between μ and λ for some classes of non-solvable groups, among them, the minimal non-solvable groups. We also provide several examples of groups not satisfying the property.
We study locally graded groups whose non-modular subgroups are soluble and satisfy some rank condition. In particular, in order to characterize locally graded groups whose subgroups are either ...modular or polycyclic, we describe (generalized) soluble groups whose non-modular subgroups are finitely generated.
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