Related papers: A note on solvable maximal subgroups in subnormal …

In this article, we study the derivations of group algebras of some important groups, namely, dihedral ($D_{2n}$), Dicyclic ($T_{4n}$) and Semi-dihedral ($SD_{8n}$). First, we explicitly classify all inner derivations of a group algebra…

Let $n$ be a non negative integer, and define $D_n$ to be the family of all finite groups having precisely $n$ conjugacy classes of nontrivial subgroups that are not self-normalizing. We are interested in studying the behavior of $D_n$ and…

A finite group $P$ is said to be \emph{primary} if $|P|=p^{a}$ for some prime $p$. We say a primary subgroup $P$ of a finite group $G$ satisfies the \emph{Frobenius normalizer condition} in $G$ if $N_{G}(P)/C_{G}(P)$ is a $p$-group provided…

In this paper we describe some properties of groups $G$ that contain a solvable subgroup of finite prime-power index (Theorem 1 and Corollaries 2--3). We prove that if $G$ is a non-solvable group that contains a solvable subgroup of index…

Dynkin's classification of maximal subalgebras of simple finite dimensional complex Lie algebras is generalized to linear Lie superalgebras. Namely, the maximal non-simple irreducible subalgebras of $\mathfrak{gl}(p|q), \mathfrak{q}(n),…

In this paper, we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or…

We say that finite groups are isospectral if they have the same sets of orders of elements. It is known that every nonsolvable finite group $G$ isospectral to a finite simple group has a unique nonabelian composition factor, that is, the…

A subgroup $H$ of a finite group $G$ is submodular in $G$ if there is a subgroup chain $H=H_0\leq\ldots\leq H_i\leq H_{i+1}\leq \ldots \leq H_n=G$ such that $H_i$ is a modular subgroup of $H_{i+1}$ for every $i$. We investigate finite…

We give a complete characterization of abelian subgroups of GL(n, R) with a locally dense (resp. dense) orbit in R^n. For finitely generated subgroups, this characterization is explicit and it is used to show that no abelian subgroup of…

We treat the problem of finding transitive subgroups G of S_n containing normal subgroups N_1 and N_2, with N_1 transitive and N_2 not transitive, such that G/N_1 is isomorphic G/N_2. We show that such G exist whenever n has a prime factor…

Let $K$ be an algebraically closed field of characteristic zero, $A= K[x_1, \dots, x_n]$ the polynomial ring in $n$ variables, and let $W_n(K)$ be the Lie algebra of all $K$-derivations of $A.$ This Lie algebra also is the free $A$-module…

For a finite group $G$, we study the probability $sp(G)$ that, given two elements $x,y \in G$, the cyclic subgroup $\langle x \rangle$ is subnormal in the subgroup $\langle x, y \rangle$. This can be seen as an intermediate invariant…

We study soluble groups G in which each subnormal subgroup H with infinite rank is commensurable with a normal subgroup, i.e. there exists a normal subgroup N such that the intersection of H and N has finite index in both H and N. We show…

For a polynomial $f = x_1^n + \dots + x_N^n$ let $G_f$ be the non--abelian maximal group of symmetries of $f$. This is a group generated by all $g \in \mathrm{GL}(N,\mathbb{C})$, rescaling and permuting the variables, so that $f(\mathbf{x})…

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 $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms, we first prove some results on the solvability of finite groups in which some maximal $A$-invariant subgroups have indices a prime or the square of a…

Denote by $G$ a finite group and let $\psi(G)$ denote the sum of element orders in $G$. In 2009, H.Amiri, S.M.Jafarian Amiri and I.M.Isaacs proved that if $|G|=n$ and $G$ is non-cyclic, then $\psi(G)<\psi(C_n)$, where $C_n$ denotes the…

We investigate the multiplication group of a special class of quasigroup called AG-group. We prove some interesting results such as: the multiplication group of an AG-group of order n is non-abelian group of order 2n and its left section is…

Assume $F$ is a finite field of order $p^f$ and $q$ is an odd prime for which $p^f-1=sq^m$, where $m \ge 1$ and $(s,q)=1$. In this article, we obtain the order of symmetric and unitary subgroup of the semisimple group algebra $FC_q.$…

Let $A$ be the ring of elements in an algebraic function field $K$ over a finite field $F_q$ which are integral outside a fixed place $\infty$. In an earlier paper we have shown that the Drinfeld modular group $G=GL_2(A)$ has automorphisms…