Narrow Results

Let $G$ be a finite group and ${ℒ}(G)$ the subgroup lattice of $G$. A subgroup $M$ of $G$ is called: (i) modular in $G$, if $M$ is a modular element (in the sense of Kurosh) of the lattice ${ℒ}(G)$; (ii) submodular in $G$ if $G$ has a chain of subgroups $M=M_0\le M_0\le \cdots \le M_t=G$, where $M_i$ is modular in $M_{i+1}$ for all $i=0,1,\dots ,t-1$. If $A$ is a subgroup of $G$, then we denote by $A_{mG}$ the subgroup of $A$, generated by all of its subgroups that are modular in $G$.

We say that a subgroup $A$ is $N$-modular in $G$ ($N\le G$), if for some modular subgroup $T$ of $G$, containing $A$, $N$avoids the pair$(T,A_{mG})$, i.e. $N\cap T=N\cap A_{mG}$.

We prove that if $G$ is a soluble finite group and each of its submodular subgroups is $N$-modular in $G$, where $N$ is the nilpotent residual of $G$, then the lattice ${ℒ}(G)$ is modular.

Let $G$ be the simple group $\mathrm{\text{PSL}}(3,2^p)$, where $p$ is a prime number. For any subgroup $H$ of $G$, we compute the Möbius function $\mu (H)$ of $H$ in the subgroup lattice of $G$. To this aim, we describe the intersections of maximal subgroups of $G$. We point out some connections of the Möbius function with other combinatorial objects, and, in this context, we compute the reduced Euler characteristic of the order complex of the subposet of $r$-subgroups of $\mathrm{\text{PGL}}(3,q)$, for any prime $r$ and any prime power $q$.

In this paper, we study the finite groups whose Chermak–Delgado measure has exactly $k$ values. We will focus especially on the case $k=2$. These groups determine an interesting class of $p$-groups containing cyclic groups of prime order and extraspecial $p$-groups.

For a finite group $G$, we associate the quantity $\beta (G)=\frac{|L(G)|}{\left|G\right|}$, where $L(G)$ is the subgroup lattice of $G$. Different properties and problems related to this ratio are studied throughout this paper. We determine the second minimum value of $\beta$ on the class of $p$-groups of order $p^n$, where $n\ge 3$ is an integer. We show that the set containing the quantities $\beta (G)$, where $G$ is a finite (abelian) group, is dense in $[0,\infty ).$ Finally, we consider $\beta$ to be a function on $L(G)$ and we indicate some of its properties, the main result being the classification of finite abelian $p$-groups $G$ satisfying $\beta (H)\le 1,\forall H\in L(G).$

This paper studies the finite groups G that have a pair (A,B) of non-trivial proper subgroups satisfying the condition that for every H ⩽ G either A ⩽ H or H ⩽ B.