International Journal of Group Theory
Volume:12 Issue: 1, Mar 2023

We study the automorphism groups of finite-dimensional cyclic Leibniz algebras. In this connection, we consider the relationships between groups, modules over associative rings and Leibniz algebras.

We completely describe the structure of locally (soluble-by-finite) groups in which all abelian subgroups are locally cyclic‎. ‎Moreover‎, ‎we prove that Engel groups with the above property are locally nilpotent‎.

We continue the investigation, that began in [M. Bianchi, A. Gillio and P. P. Pálfy, A note on finite groups in which the conjugacy class sizes form an arithmetic progression, Ischia group theory 2010, World Sci. Publ., Hackensack, NJ (2012) 20--25.] and [M. Bianchi, S. P. Glasby and Cheryl E. Praeger, Conjugacy class sizes in arithmetic progression, J. Group Theory, 23 no. 6 (2020) 1039--1056.], into finite groups whose set of nontrivial conjugacy class sizes form an arithmetic progression. Let $G$ be a finite group and denote the set of conjugacy class sizes of $G$ by ${\rm cs}(G)$. Finite groups satisfying ${\rm cs}(G) = \{1, 2, 4, 6\}$ and $\{1, 2, 4, 6, 8\}$ are classified in [M. Bianchi, S. P. Glasby and Cheryl E. Praeger, Conjugacy class sizes in arithmetic progression, J. Group Theory, 23 no. 6 (2020) 1039--1056.] and [M. Bianchi, A. Gillio and P. P. Pálfy, A note on finite groups in which the conjugacy class sizes form an arithmetic progression, Ischia group theory 2010, World Sci. Publ., Hackensack, NJ (2012) 20--25.], respectively, we demonstrate these examples are rather special by proving the following. There exists a finite group $G$ such that ${\rm cs}(G) = \{1, 2^{\alpha}, 2^{\alpha+1}, 2^{\alpha}3 \}$ if and only if $\alpha =1$. Furthermore, there exists a finite group $G$ such that ${\rm cs}(G) = \{1, 2^{\alpha}, 2^{\alpha +1}, 2^{\alpha}3, 2^{\alpha +2}\}$ and $\alpha$ is odd if and only if $\alpha=1$.

In this survey we shall describe some recent criteria for solvability, nilpotency and other properties of finite groups $G$, based either on the orders of the elements of $G$ or on the orders of the subgroups of $G$.

A group $G$ is existentially closed (algebraically closed) if every finite system of equations and in-equations that has coefficients in $G$ and has a solution in an overgroup $H\geq G$ has a solution in $G$. Existentially closed groups were introduced by W. R. Scott in 1951. B. H. Neumann posed the following question in 1973: Does there exist explicit examples of existentially closed groups? Generalized version of this question is as follows: Let $\kappa$ be an infinite cardinal. Does there exist explicit examples of $\kappa$-existentially closed groups? Recently an affirmative answer was given to Neumann's question and the generalized version of it, by Kaya-Kegel-Kuzucuo\u{g}lu. We give a survey of these results. We also prove that, there are maximal subgroups of $\kappa$-existentially existentially closed groups and provide some information about subgroups containing the centralizer of subgroups generated by fewer than $\kappa$-elements. This generalizes a result of Hickin-Macintyre.