A subgroup $H$ of a group $G$ is said to be an $ICΦ$-subgroup of $G$ if $H \cap [H,G] \le Φ(H)$. We analyze the structure of a finite group $G$ under the assumption that some given subgroups of $G$ are $ICΦ$-subgroups of $G$. A new characterization of finite abelian groups and some new criteria for $2$-nilpotence and nilpotence of finite groups will be obtained. Moreover, we will obtain two criteria for a finite group to lie in a given solvably saturated formation containing the class of finite supersolvable groups.
We give a description of a finite group whose maximal subgroups possess only soluble proper subgroups, which implies the answer to the well-known question on composition factors of finite groups, whose second maximal subgroups are soluble.
It is known that in any free group the isolator of finitely generated subgroup is finitely generated subgroup. A very simple proof of this statement is proposed.
We consider abelain subgroups of small index in finite groups. More generally, we consider subgroups such that the product of their index by the index of their centralizer is small.
We first give complete characterizations of the structure of finite group $G$ in which every subgroup (or non-nilpotent subgroup, or non-abelian subgroup) is a TI-subgroup or subnormal or has $p'$-order for a fixed prime divisor $p$ of $|G|$. Furthermore, we prove that every self-centralizing subgroup (or non-nilpotent subgroup, or non-abelian subgroup) of $G$ is a TI-subgroup or subnormal or has $p'$-order for a fixed prime divisor $p$ of $|G|$ if and only if every subgroup (or non-nilpotent subgroup, or non-abelian subgroup) of $G$ is a TI-subgroup or subnormal or has $p'$-order. Based on th…
Supersolubility of a finite group $G=\langle A,B\rangle$ with the nilpotent derived subgroup $G^\prime$ is established under the condition that the subgroups $A$ and $B$ are both subnormal and supersoluble.
Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. We say that $H$ is s-semipermutable in $G$ if $HG_p = G_pH$ for any Sylow $p$-subgroup $G_p$ of $G$ with $(p, |H|) = 1$. We investigate the influence of s-semipermutable subgroups on the structure of finite groups. Some recent results are generalized.
Given a finite group $G$, we denote by $L(G)$ the subgroup lattice of $G$ and by ${\rm Isolated}(G)$ the set of isolated subgroups of $G$. In this note, we describe finite groups $G$ such that $|{\rm Isolated}(G)|=|L(G)|-k$, where $k=0,1,2$.
Tărnăuceanu described the finite groups $G$ having exactly $|G|-1$ cyclic subgroups. In "Finite Groups with a Prescribed Number of Cyclic Subgroups,", we used elementary methods to completely characterize those finite groups $G$ having exactly $|G|-Δ$ cyclic subgroups for $Δ=2, 3, 4$ and $5$. In this paper, we prove that for any $Δ>0$ if $G$ has exactly $|G|-Δ$ cyclic subgroups, then $|G|\le 8Δ$ and therefore the number of such $G$ is finite. We then use the computer program GAP to find all $G$ with exactly $|G|-Δ$ cyclic subgroups for $Δ=1,\ldots,32$.
The paper is concerned with Kropholler's conjecture on splitting a finitely generated group over a codimension-1 subgroup. For a subgroup H of a group G, we define the notion of "finite splitting height" which generalises the finite-height property. By considering the dual CAT(0) cube complex associated to a codimension-1 subgroup H in G, we show that the Kropholler-Roller conjecture holds when H has finite splitting height in G. Examples of subgroups of finite height are stable subgroups or more generally strongly quasiconvex subgroups. Examples of subgroups of finite splitting height include…
Given a prime power $p^d$ with $p$ a prime and $d$ a positive integer, we classify the finite groups $G$ with $p^{2d}$ dividing $|G|$ in which all subgroups of order $p^d$ are complemented and the finite groups $G$ having a normal elementary abelian Sylow $p$-subgroup $P$ such that $p^d<|P|$ in which all subgroups of order $p^d$ are complemented.
We say that a subgroup $H$ is isolated in a group $G$ if for every $x\in G$ we have either $x\in H$ or $\langle x\rangle\cap H=1$. In this short note, we describe the set of isolated subgroups of a finite abelian group. The technique used is based on an interesting connection between isolated subgroups and the function sum of element orders of a finite group.
