Related papers: On finite groups with given $IC\Phi$-subgroups

Given a finite group $G$, we denote by $\nu(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup. We prove that if $\nu(G)>1/12,$ then $G$ is solvable.

Suppose that $G$ is a finite group and $H$ is a nilpotent subgroup of $G$. If a character of $H$ induces an irreducible character of $G$, then the generalized Fitting subgroup of $G$ is nilpotent.

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…

In the present paper, the structure of a finite group $G$ having a nonnormal T.I. subgroup $H$ which is also a Hall $\pi$-subgroup is studied. As a generalization of a result due to Gow, we prove that $H$ is a Frobenius complement whenever…

Let G be a profinite group. The following results are proved. The commutator subgroup G' is finite if and only if G is covered by countably many abelian subgroups. The group G is finite-by-nilpotent if and only if G is covered by countably…

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.

The number of subgroups and the number of cyclic subgroups are natural combinatorial invariants of a finite group. We investigate how restrictions on these quantities, together with the number of distinct prime divisors of $|G|$, enforce…

Given a finite group $G,$ we denote by $\Delta(G)$ the graph whose vertices are the proper subgroups of $G$ and in which two vertices $H$ and $K$ are joined by an edge if and only if $G=\langle H,K\rangle.$ We prove that if there exists a…

Let $G$ be a finite group and let $\pi$ be a set of primes. In this paper, we prove a criterion for the existence of a solvable $\pi$-Hall subgroup of $G$, precisely, the group $G$ has a solvable $\pi$-Hall subgroup if, and only if, $G$ has…

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies $ \mathscr L $-$ \Pi $-property in $ G $ if $ | G / K : N _{G / K} (HK/K)| $ is a $ \pi (HK/K) $-number for all maximal $ G $-invariant subgroup $ K $ of $ H^{G}…

In this paper, we discuss about finite groups in which, CGH = NGH, for every abelian subgroup H of non prime power order. Also, we classify all such nilpotent and minimal non nilpotent groups.

In this paper, we investigate the structure of finite group G by assuming that the intersections between p-sylowizers of some p-subgroups of G and $O^p(G)$ are S-permutable in G. We obtain some criterions for p-nilpotency of a finite group.

A finite group $G$ is said to be a $\mathcal{B}_{\psi}$-group if $\psi(H)<|G|$ for any proper subgroup $H$ of $G$, where $\psi(H)$ denotes the sum of element orders of $H$. In this paper, we characterize the $\mathcal{B}_{\psi}$-groups up…

In this paper, we provide new criteria for the solvability and supersolvability of a finite group based on its number of cyclic subgroups. A finite group G is called n-cyclic if it contains n cyclic subgroups. This paper also partially…

Let $\mathfrak{F}$ be a formation and $G$ a finite group. A subgroup $H$ of $G$ is said to be weakly $\mathfrak{F}_{s}$-quasinormal in $G$ if $G$ has an $S$-quasinormal subgroup $T$ such that $HT$ is $S$-quasinormal in $G$ and $(H\cap…

For any finite group Q not of prime power order, we construct a group G that is virtually of type F, contains infinitely many conjugacy classes of subgroups isomorphic to Q, and contains only finitely many conjugacy classes of other finite…

Let $\sigma =\{\sigma_i |i\in I\}$ is some partition of all primes $\mathbb{P}$ and $G$ a finite group. A subgroup $H$ of $G$ is said to be $\sigma$-subnormal in $G$ if there exists a subgroup chain $H=H_0\leq H_1\leq \cdots \leq H_n=G$…

Let $p$ be a prime, $S$ be a $p$-group and $\mathcal{F}$ be a saturated fusion system over $S$. Then $\mathcal{F}$ is said to be supersolvable, if there exists a series of $S$, namely $1 = S_0 \leq S_1 \leq \cdots \leq S_n = S$, such that…

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $\sigma $-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall…

Necessary and sufficient conditions for finite commutative semihypergroups to be built from abelian groups of the same order are established.