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We characterise finite unitary rings $R$ such that all Sylow subgroups of the group of units $R^{\ast }$ are cyclic. To be precise, we show that, up to isomorphism, $R$ is one of the three types of rings in $\{O,E,O\oplus E\}$ , where $O\in \{GF(q),\mathbb{Z}_{p^{\unicode[STIX]{x1D6FC}}}\}$ is a ring of odd cardinality and $E$ is a ring of cardinality $2^{n}$ which is one of seven explicitly described types.

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 [Formula: see text] and [Formula: see text] is a ring of cardinality [Formula: see text] which is one of the six explicitly described types.

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 automorphism [Formula: see text] such that [Formula: see text] and [Formula: see text] is the largest normal [Formula: see text]-subgroup of [Formula: see text].

This paper aimed to study distributivity and Jordan-Dedekind condition in hyperlattices. Hyperlattice L satisfies the Jordan-Dedekind condition if all maximal chains between the same end points hav...

In this paper, we introduce the notions of strong ” ∧ ”-hyperlattices, hyperideals and hyperfilters in strong ”∧”-hyperlattices. Also, we give equivalence conditions for prime hyperfilters (hyperideal) in strong ∧-hyperlattices. Distributivity (dual distributivity) in ”∧”-hyperlattices,IA-hyperideals and prime hyperfilters in strong ” ∧ ”-hyperlattices are investigated.

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(G) when GZ(G) is minimal nonabelian and this generalizes some previous results. We give a necessary and sufficient condition for a primitive n-centralizer group G with the minimal nonabelian central factor. Also we show that if GZ(G)≅S4, then G is a primitive 14-centralizer group and ω(G) = 10 or 13. Finally we confirm Conjecture 2.4 in [A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq. 7(2) (2000), 139-146].

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).

Given a finite group G, we denote by $$\psi (G)$$ 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 $$\psi (G)=1+t|G|$$ 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 second maximum sum of the element orders among all finite groups of the same square-free order.

We characterize the structure of finite unitary rings [Formula: see text] in which every proper subring is commutative and show that [Formula: see text] is a commutative ring.

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 $R\cong M_2(GF(2))\oplus A$ where $M_2(GF(2))$ is the ring of $2\times 2$ matrices over the finite field $GF(2)$ and $A$ is a direct sum of finite field $GF(2)$., Comment: 10 pages

Let $G$ be a finite group. Then we denote $\psi(G) = \sum_{x\in G}o(x)$ 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., Comment: 9 pages

Let G be a finite group. Then we denote ψ(G) the sum of element orders in G. In [1] it is proved that if G is a non-cyclic group of order n, then ψ(G) < ψ(C n ), where C n is the cyclic group of order n. Here we generalize and improve this result, and also we give an application of this improvement.

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].

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]. In this paper, we characterize all groups [Formula: see text] such that [Formula: see text] and [Formula: see text] is not a 2-group.

We show that if [Formula: see text] is a non-central Lie ideal of a ring [Formula: see text] with Char[Formula: see text], such that all of its nonzero elements are invertible, then [Formula: see text] is a division ring. We prove that if [Formula: see text] is an [Formula: see text]-central algebra and [Formula: see text] is a Lie ideal without zero divisor such that the set of multiplicative cosets [Formula: see text] is of finite cardinality, then either [Formula: see text] is a field or [Formula: see text] is central. We show the only non-central Lie ideal without zero divisor of a non-commutative central [Formula: see text]-algebra [Formula: see text] with Char[Formula: see text] and radical over the center is [Formula: see text], the additive commutator subgroup of [Formula: see text] and in this case [Formula: see text] is a generalized quaternion algebra. Finally we prove that if [Formula: see text] is a Lie ideal without zero divisor in a central [Formula: see text]-algebra with characteristic not 2 and if [Formula: see text] is a finite residual group, then [Formula: see text] is central.

Cartan–Brauer–Hua Theorem is a well-known theorem which states that if [Formula: see text] is a subdivision ring of a division ring [Formula: see text] which is invariant under all elements of [Formula: see text] or [Formula: see text] for all [Formula: see text], then either [Formula: see text] or [Formula: see text] is contained in the center of [Formula: see text]. The invariance idea of this basic theorem is the main notion of this paper. We prove that if [Formula: see text] is a division ring with involution [Formula: see text] and [Formula: see text] is a subspace of [Formula: see text] which is invariant under all symmetric elements of [Formula: see text], then either [Formula: see text] is contained in the center of [Formula: see text] or is a Lie ideal of [Formula: see text]. Also, we show that if [Formula: see text] is a self-invariant subfield of a non-commutative division ring [Formula: see text] with a nontrivial automorphism, then [Formula: see text] contains at least one non-central proper subfield of [Formula: see text].