Mohsen Amiri

Federal University of Amazonas | UFAM · Department of Mathematics

Let [Formula: see text] be a finite solvable group and [Formula: see text] a non-normal core-free solvable subgroup of [Formula: see text]. We show that if the normalizer of any nontrivial normal subgroup of [Formula: see text] is equal to [Formula: see text], then [Formula: see text] has a nilpotent normal complement [Formula: see text] such that...

As a natural continuation of study $LCM$-groups, we explore other properties of $LCM$-groups and $LC$-series. We obtain some characterizations of finite groups which are not LCM-groups but all proper sections are $LCM$-groups. Also, for a $p$-group $G$, we prove that $G$ is a $LC$-nilpotent group and we obtain a bound for its $LC$-nilpotency. Final...

Let $R$ be a finite unitary ring with $R^*$ a non-solvable group, but such that for all proper unitary subrings $S$ of $R$, $S^*$ is a solvable group. We classify such rings. Moreover we show that any ring with a non-solvable group of units and of order $p^n$, $n\leq 5$, is of that form and we also analyze the structure of rings of order $p^6$.

Let G be a periodic group, and let LCM(G) be the set of all x∈G such that o(xnz) divides the least common multiple of o(xn) and o(z) for all z in G and all integers n. In this paper, we prove that the subgroup generated by LCM(G) is a locally nilpotent characteristic subgroup of G whenever G is a locally finite group.

Let [Formula: see text] be a unitary ring of finite cardinality [Formula: see text], where [Formula: see text] is a prime number and [Formula: see text]. We show that if the group of units of [Formula: see text] has at most one subgroup of order [Formula: see text], then [Formula: see text] where [Formula: see text] is a finite ring of order [Formu...

Let [Formula: see text] be a finite unitary ring such that [Formula: see text], where [Formula: see text] is the prime ring and [Formula: see text] is not a nilpotent group. We show that if all proper subgroups of [Formula: see text] are nilpotent groups, then the cardinality of [Formula: see text] is a power of 2. In addition, if [Formula: see tex...

Let R be a unitary ring of finite cardinality pk, where p is a prime and p k. We show that if the group of units of R has at most one subgroup of order p, then, B where B is a finite ring of order k and A is a ring of cardinality p which is one of six explicitly described types.

We characterize finite unitary rings $R$ such that all Sylow subgroups of the group of units $R^*$ are cyclic. To be precize, we show that, up to isomorphism, $R$ is one of the three types of rings in $ \{O, E, O\bigoplus E\}$, where $O \in \{ GF(q), {\mathbb{Z}}_{p^{\alpha}} \}$ is a ring of odd cardinality and either $$ E \in \{ M_2(GF(2)), T_2(G...

This paper generalizes the concept of element order and studies some of its basic properties. By employing these latter we prove two solvability criteria for finite groups as well as the following result. Given a finite group $G$ of order $n$ then there exists a bijection $f$ from $G$ onto a cyclic group of order $n$ such that for each element $x\i...

Let $R$ be a finite unitary ring such that $R=R_0[R^*]$ where $R_0$ is the prime ring and $R^*$ is not a nilpotent group. We show that if all proper subgroups of $R^*$ are nilpotent groups, then the cardinal of $R$ is a power of prime number 2. In addition, if $(R/Jac(R))^*$ is not a $p-$group, then either $R\cong M_2(GF(2))$ or $R\cong M_2(GF(2))\...

Let [Formula: see text] be a finite group and [Formula: see text] be the set of the elements [Formula: see text] of [Formula: see text] such that [Formula: see text] where [Formula: see text]. In this paper, we give strong restrictions on the structure of the quotients [Formula: see text] where [Formula: see text] is a finite group with an automorp...

For a finite group G, let |Cent(G)| and ω(G) denote the number of centralizers of its elements and the maximum size of a set of pairwise noncommuting elements of it, respectively. A group G is called n-centralizer if |Cent(G)| = n and primitive n-centralizer if |Cent(G)|=|Cent(GZ(G))|=n . In this paper, among other results, we find |Cent(G)| and ω...

Let G be a finite solvable group of order n and p be a prime divisor of n. In this article, we prove that if the Sylow p-subgroup of G is neither cyclic nor generalized quaternion, then there exists a bijection f from G onto the abelian group Cnp×Cp such that for each x ∈ G, the order of x divides the order of f(x).

For a finite group G, let Cent(G) denote the set of centralizers of single elements of G. In this note we prove that if |G|≤3/2|cent(G)| and G is 2-nilpotent, then G (Formula Present)S3;D10 or S3×S3. This result gives a partial and positive answer to a conjecture raised by A. R. Ashrafi [On Finite groups with a given number of centralizers, Algebra...

Let G be a finite group and I(G) be the set of the elements x of G such that x2 = 1. In this paper, we characterize all groups G such that |I(G)|≥ |G| 3 and G is not a 2-group.

We show that if I is a non-central Lie ideal of a ring R with Char(R) ≠ 2, such that all of its nonzero elements are invertible, then R is a division ring. We prove that if R is an F-central algebra and I is a Lie ideal without zero divisor such that the set of multiplicative cosets {aF | a ∈ I} is of finite cardinality, then either R is a field or...

We show that if I is a non-central Lie ideal of a ring R with Char(R) ̸= 2, such that all of its non-zero elements are invertible, then R is a division ring. We prove that if R is an F-central algebra and I is a Lie ideal without zero divisor such that the set of multiplicative cosets faF j a 2 Ig is of finite cardinality, then either R is a field...

Cartan-Brauer-Hua Theorem is a well known theorem which states that if R is a subdivision ring of a division ring D which is invariant under all elements of D or dRd􀀀1 � R for all d 2 D n f0g, then either R = D or R is contained in the center of D. The invariance idea of this basic theorem is the main notion of this paper. We prove that if D is a d...

Let G be a finite group. Then we denote psi(G) = Sigma(x is an element of G)o(x) where o(x) is the order of the element x in G. In this paper we characterize some finite p-groups (p a prime) by psi, and their orders.

Given a finite group G, we denote by ψ(G) the sum of the element orders in G. In this article, we prove that if t is the number of nonidentity conjugacy classes in G, then ψ(G) = 1 + t| G| if and only if G is either a group of prime order or a nonabelian group of the square-free order with two prime divisors. Also we find a unique group with the se...

We give a new proof of the well known Wedderburn's little theorem (1905) that a finite division ring is commutative. We apply the concept of Frobenius kernel in Frobenius representation theorem in finite group theory to build a proof.

Let G be a finite group. Then we denote ψ(G) the sum of element orders in G. In [11. Amiri , H. , Jafarian Amiri , S. M. , Isaacs , I. M. ( 2009 ). Sums of element orders in finite groups . Comm. Algebra 37 ( 9 ): 2978 – 2980 . [Taylor & Francis Online], [Web of Science ®]View all references] it is proved that if G is a non-cyclic group of order...

Amiri, Jafarian Amiri and Isaacs proved that the cyclic group has maximum sum of element orders on all groups of the same order. In this article we characterize finite groups which have maximum sum of element orders among all noncyclic groups of the same order. This result confirms the conjecture posed in Jafarian Amiri (2013) [2].

We give a new proof of the well known Wedderburn's little theorem (1905) that a �nite division ring is commutative. We apply the concept of Frobenius kernel in Frobenius representation theorem in �nite group theory to build a proof. Keywords: Division ring, maximal sub�eld, Frobenius representation theorem.