The complete classification of the finite simple groups that are $(2,3)$-generated is a problem which is still open only for orthogonal groups. Here, we construct $(2, 3)$-generators for the finite odd-dimensional orthogonal groups $\Omega _{2k+1}(q)$, $k\geq 4$. As a byproduct, we also obtain $(2,3)$-generators for $\Omega _{4k}^+(q)$ with $k\geq 3$ and q odd, and for $\Omega _{4k+2}^\pm (q)$ with $k\geq 4$ and $q\equiv \pm 1~ \mathrm {(mod~ 4)}$.

The codegree of an irreducible character $\chi $ of a finite group G is $|G : \ker \chi |/\chi (1)$. We show that the Ree group ${}^2G_2(q)$, where $q = 3^{2f+1}$, is determined up to isomorphism by its set of codegrees.

In this paper, we consider the $T$- and $V$-versions, ${T_\tau }$ and ${V_\tau }$, of the irrational slope Thompson group ${F_\tau }$ considered in J. Burillo, B. Nucinkis and L. Reeves [An irrational-slope Thompson's group, Publ. Mat. 65 (2021), 809–839]. We give infinite presentations for these groups and show how they can be represented by tree-pair diagrams similar to those for $T$ and $V$. We also show that ${T_\tau }$ and ${V_\tau }$ have index-$2$ normal subgroups, unlike their original Thompson counterparts $T$ and $V$. These index-$2$ subgroups are shown to be simple.

Let G be a finite group, let ${\mathrm{Irr}}(G)$ be the set of all irreducible complex characters of G and let $\chi \in {\mathrm{Irr}}(G)$ . Define the codegrees, ${\mathrm{cod}}(\chi ) = |G: {\mathrm{ker}}\chi |/\chi (1)$ and ${\mathrm{cod}}(G) = \{{\mathrm{cod}}(\chi ) \mid \chi \in {\mathrm{Irr}}(G)\} $ . We show that the simple group ${\mathrm{PSL}}(2,q)$ , for a prime power $q>3$ , is uniquely determined by the set of its codegrees.

We show that any simple group of Morley rank 5 is a bad group all of whose proper definable connected subgroups are nilpotent of rank at most 2. The main result is then used to catalog the nonsoluble connected groups of Morley rank 5.

Let $T$ be a finite simple group of Lie type in characteristic $p$, and let $S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is a Sylow $p$-subgroup except for an explicit list of exceptions and that $S$ is always ‘large’ in the sense that $|T|^{1/3}<|S|\leq |T|^{1/2}$. One might anticipate that, moreover, the Sylow $r$-subgroups of $T$ with $r\neq p$ are usually significantly smaller than $S$. We verify this hypothesis by proving that, for every $T$ and every prime divisor $r$ of $|T|$ with $r\neq p$, the order of the Sylow $r$-subgroup of $T$ is at most $|T|^{2\lfloor \log _{r}(4(\ell +1)r)\rfloor /\ell }=|T|^{O(\log _{r}(\ell )/\ell )}$, where $\ell$ is the Lie rank of $T$.

Let $T$ be a locally finite tree without vertices of degree $1$. We show that among the closed subgroups of $\text{Aut}(T)$ acting with a bounded number of orbits, the Chabauty-closure of the set of topologically simple groups is the set of groups without proper open subgroup of finite index. Moreover, if all vertices of $T$ have degree ${\geqslant}3$, then the set of isomorphism classes of topologically simple closed subgroups of $\text{Aut}(T)$ acting doubly transitively on $\unicode[STIX]{x2202}T$ carries a natural compact Hausdorff topology inherited from Chabauty. Some of our considerations are valid in the context of automorphism groups of locally finite connected graphs. Applications to Weyl-transitive automorphism groups of buildings are also presented.

We show that any simple group of Morley rank 4 must be a bad group with no proper definable subgroups of rank larger than 1. We also give an application to groups acting on sets of Morley rank 2.

This paper is a brief survey of recent results and some open problems related to linear groups of finite Morley rank, an area of research where Bruno Poizat's impact is very prominent. As a sign of respect to his strongly expressed views that mathematics has to be done, written and pulished only in the native tongue of the immediate author–the scribe, in effect–of the text, I insist on writing my paper in Russian, even if the results presented belong to a small but multilingual community of researchers of American, British, French, German, Kazakh, Russian, Turkish origin. To emphasise even further the linguistic subtleties involved, I use British spelling in the English fragments of my text.

We characterise the class of finite solvable groups by two-variable identities in a way similar to the characterisation of finite nilpotent groups by Engel identities. Let $u_1=x^{-2}y\min x$, and $u_{n+1} = [xu_nx\min,yu_ny\min]$. The main result states that a finite group G is solvable if and only if for some n the identity $u_n(x,y)\equiv 1$ holds in G. We also develop a new method to study equations in the Suzuki groups. We believe that, in addition to the main result, the method of proof is of independent interest: it involves surprisingly diverse and deep methods from algebraic and arithmetic geometry, topology, group theory, and computer algebra (SINGULAR and MAGMA).

Order components of a finite simple group were introduced in [4]. It was proved that some non-abelian simple groups are uniquely determined by their order components. As the main result of this paper, we show that groups $PS{{U}_{11}}(q)$ are also uniquely determined by their order components. As corollaries of this result, the validity of a conjecture of J. G. Thompson and a conjecture of W. Shi and J. Bi both on $PS{{U}_{11}}(q)$ are obtained.