Related papers: On the maximal subgroups of almost simple and prim…

Let G be a finite group and S a subset of G\{0}. We call S an additive basis of G if every element of G can be expressed as a sum over a nonempty subset in some order. Let cr(G) be the smallest integer t such that every subset of G\{0} of…

Let G be a connected linear algebraic group over an algebraically closed field k, and let H be a connected closed subgroup of G. We prove that the homogeneous variety G/H is a rational variety over k whenever H is solvable, or when dim(G/H)…

We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite…

Let $G$ be a finite almost simple group with socle $G_0$. In this paper we prove that whenever $G/G_0$ is abelian, then there exists an abelian subgroup $A$ of $G$ such that $G=AG_0$. We propose a few applications of this structural…

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. $G$ is said to be \emph{$\sigma$-soluble} if every chief factor $H/K$ of $G$ is a $\sigma_{i}$-group for some $i=i(H/K)$. A…

Let $o(G)$ be the average order of a finite group $G$. In this paper, we prove that if $o(G)<\frac{31}{12}$\,, then $G$ is supersolvable. Moreover, we have $o(G)=\frac{31}{12}$ if and only if $G\cong A_4$. We also classify finite groups $G$…

A 2-covering for a finite group $G$ is a set of proper subgroups of $G$ such that every pair of elements of $G$ is contained in at least one subgroup in the set. The minimal number of subgroups needed to 2-cover a group $G$ is called the…

Let $n>0$ be an integer and $\mathcal{X}$ be a class of groups. We say that a group $G$ satisfies the condition $(\mathcal{X},n)$ whenever in every subset with $n+1$ elements of $G$ there exist distinct elements $x,y$ such that $<x,y>$ is…

We prove that there exists a constant $k$ with the property: if $\calC$ is a conjugacy class of a finite group $G$ such that every $k$ elements of $\calC$\ generate a solvable subgroup then $\calC$ generates a solvable subgroup. In…

A subgroup H of a group G is called inert if for each $g\in G$ the index of $H\cap H^g$ in $H$ is finite. We give a classification of soluble-by-finite groups $G$ in which subnormal subgroups are inert in the cases where $G$ has no…

A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code in $\Gamma$ if every vertex of $\Gamma$ is at distance no more than $1$ to exactly one vertex of $C$. A subset $C$ of a group $G$ is called a perfect code of $G$ if…

We study finite groups $G$ with the property that for any subgroup $M$ maximal in $G$ whose order is divisible by all the prime divisors of $|G|$, $M$ is supersolvable. We show that any nonabelian simple group can occur as a composition…

Let $G$ be a finite abelian group and $s$ be a positive integer. A subset $A$ of $G$ is called a {\em perfect $s$-basis of $G$} if each element of $G$ can be written uniquely as the sum of at most $s$ (not-necessarily-distinct) elements of…

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies partial $\Pi$-property in $G$ if there exists a chief series $\mathit{\Gamma}_G:1=G_0<G_1<\cdots<G_n=G$ of $G$ such that for every $G$-chief factor $G_i/G_{i-1}$ ($1\leq…

Let A be a subset of an abelian group G. We say that A is sum-free if there do not exist x,y and z in A satisfying x + y = z. We determine, for any G, the cardinality of the largest sum-free subset of G. This equals c(G)|G| where c(G) is a…

A subset $C$ of the vertex set $V$ of a graph $\Gamma$ is called a perfect code in $\Gamma$ if every vertex in $V\setminus C$ is adjacent to exactly one vertex in $C$. Given a group $G$ and a subgroup $H$ of $G$, a subgroup $A$ of $G$…

M.Newman has asked if it is the case that whenever H and K are isomorphic subgroups of a finite solvable group G with H maximal, then K is also maximal. This question was considered in a paper of I.M. Isaacs and the second author, where…

We determine all maximal subgroups of the almost simple groups with socle $T=\PSL(2,q)$, that is, of all groups $G$ such that $\PSL(2,q)\leqslant G\leqslant\PGammaL(2,q)$, with $q\geq 4$.

The generalized order $e_G(g)$ of an element $g$ of a group $G$ is the smallest positive integer $k$ such that there exist $x_1,\ldots,x_k \in G$ such that $g^{x_1} \ldots g^{x_k}=1$, where $g^x=x^{-1}gx$. Let $e(G) = \max \{e_G(g)\ |\ g…

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}…