On sublattices of the subgroup lattice defined by formation Fitting sets

Throughout this paper, all groups are finite. Let σ={σi|i∈I} be some partition of the set of all primes P. If n is an integer, the symbol σ(n) denotes the set {σi|σi∩π(n)≠∅}; σ(G)=σ(|G|) and σ(F)=∪...

On n-multiply σ-local formations of finite groups

G-covering subgroup systems for some classes of σ-soluble groups

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.

/pdf/on-some-classes-of-sublattices-of-the-subgroup-lattice-503zmnmwej.pdf

On some classes of sublattices of the subgroup lattice

Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at primes above $p$ In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over $\mathbb{Z}_p^2$-extensions of an imaginary quadratic field where $p$ splits completely We only discuss the algebraic aspects of these objects through Iwasawa theory We also attempt to give some of the recent results implying the vanishing of the $\mu$-invariant under the hypothesis of Conjecture A Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case We highlight properties of signed Selmer groups (when $E$ has good supersingular reduction) which are completely analogous to the classical Selmer group (when $E$ has good ordinary reduction) In this survey paper, we do not present any proofs, however we have tried to give references of the discussed results for the interested reader

Conjecture A and $\mu$-invariant for Selmer groups of supersingular elliptic curves

On σ-supersoluble groups and one generalization of CLT-groups

Let $\\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\\mathcal{L}}_{\\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\\in \\mathfrak{F}$. A chief factor $H/K$ of $G$ is $\\mathfrak{F}$-central in $G$ if $(H/K)\\rtimes (G/C_{G}(H/K))\\in \\mathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $\\mathfrak{F}$-central in $G$ for every subgroup $A\\in {\\mathcal{L}}_{\\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a PST-group if and only if $A^{G}/A_{G}\\leq Z_{\\infty }(G/A_{G})$ for every subgroup $A\\in {\\mathcal{L}}_{\\mathfrak{N}}(G)$, where $\\mathfrak{N}$ is the class of all nilpotent groups.

On a lattice characterisation of finite soluble $PST$-groups

Throughout this paper, all groups are finite. Let $\sigma =\{\sigma_{i} | i\in I \}$ be some partition of the set of all primes $\Bbb{P}$. If 
$n$ is an integer, the symbol $\sigma (n)$ denotes 
the set $\{\sigma_{i} |\sigma_{i}\cap \pi (n)\ne 
\emptyset \}$. The integers $n$ and $m$ are called $\sigma$-coprime if $\sigma (n)\cap \sigma (m)=\emptyset$. 
Let $t > 1$ be a natural number and let $\mathfrak{F}$ be a class of groups. Then we say that $\mathfrak{F}$ is $\Sigma_{t}^{\sigma}$-closed provided $\mathfrak{F}$ contains each group 
$G$ with subgroups $A_{1}, \ldots , A_{t}\in 
\mathfrak{F}$ whose indices $|G:A_{1}|$, $\ldots$, $|G:A_{t}|$ are pairwise $\sigma$-coprime. 
In this paper, we study $\Sigma_{t}^{\sigma}$-closed classes of finite groups.

One application of the $\sigma$-local formations of finite groups

Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\cal L}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$-central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K)) \in\mathfrak{F}$. 
We study the structure of $G$ under the hypothesis that every chief factor 
of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$-central in $G$ for every subgroup $A\in {\cal L}_{\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a $PST$-group if and only if $A^{G}/A_{G}\leq Z_{\infty}(G/A_{G})$ for every subgroup $A\in {\cal L}_{\mathfrak{N}}(G)$, where $\mathfrak{N}$ is the class of all nilpotent groups.

Zhang Chi

Papers

On σ-supersoluble groups and one generalization of CLT-groups

On n-multiply σ-local formations of finite groups

On a lattice characterisation of finite soluble \(PST\)-groups

One application of the $\sigma$-local formations of finite groups

On a lattice characterization of finite soluble $PST$-groups