Related papers: A characterisation of almost simple groups with so…

We identify the sporadic groups M(23) and F_2 from the approximate structure of the centralizer of an element of order 3.

We identify the groups $PSU_6(2)$, $PSU_6(2):2$, $PSU_6(2):3$ and $Aut(PSU_6(2))$ from the structure of the centralizer of an element of order 3.

We identify the sporadic simple group $\mathrm{M}_{12}$ and the simple group $\mathrm{SL}_3(3)$ from some part of their 3-local structure and give a graph theoretic analogue of the resulting theorem.

The spectrum of a finite group is the set of its elements orders. Groups are said to be isospectral if their spectra coincide. For every finite simple exceptional group $L=E_7(q)$, we prove that each finite group isospectral to $L$ is…

We prove that infinite definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. Then, we identify conditions on automorphisms of a stable group that make it resemble…

Among the simplest invariants of the sporadic finite simple groups are their outer automorphism groups. For 12 of the 26 possible isomorphism types of a sporadic simple group G, the outer automorphism group Out(G) has order 2, and in the…

We first show that every quasisimple sporadic group possesses an unmixed strongly real Beauville structure aside from the Mathieu groups M11 and M23 (and possibly 2B and M). We go on to show that no almost simple sporadic group possesses a…

We say that a finite almost simple $G$ with socle $S$ is admissible (with respect to the spectrum) if $G$ and $S$ have the same sets of orders of elements. Let $L$ be a finite simple linear or unitary group of dimension at least three over…

The spectrum of a finite group is the set of its element orders, and two groups are said to be isospectral if they have the same spectra. A finite group $G$ is said to be recognizable by spectrum, if every finite group isospectral with $G$…

We answer in the negative a question of Hartley about representations of finite groups, by constructing examples of finite simple groups with arbitrarily large representations whose endomorphism ring consists of just the scalars. We show as…

In this article we give self-contained existence proofs for the sporadic simple groups Co_2 and Fi_{22} using the second author's algorithm [10] constructing finite simple groups from irreducible subgroups of GL_n(2). These two sporadic…

In adjoint reductive groups $H$ of type $\mathsf{D}$ we show that for every semisimple element $s$, its centralizer splits over its connected component, i.e., $C_H(s) = C_H(s)^\circ \rtimes \check A$ for some complement $\check A$ with…

This note gives an elementary proof that the symmetric groups possess only one exceptional symmetry. I am referring to the fact that the outer automorphism group of the symmetric group $S_n$ is trivial unless $n=6$ and the outer…

In this PhD thesis we discuss methods of recognizing finite groups by the structure of normalizers of certain 3-subgroups. We explain a method for characterizing groups using character theoretic and block theoretic methods and we use these…

We give a new computer-assisted proof of the classification of maximal subgroups of the simple group ${}^2E_6(2)$ and its extensions by any subgroup of the outer automorphism group $S_3$. This is not a new result, but no earlier proof…

The purpose of this paper is to explore the concept of localization, which comes from homotopy theory, in the context of finite simple groups. We give an easy criterion for a finite simple group to be a localization of some simple subgroup…

We provide 3-local characterizations of the Harada--Norton sporadic simple group and its automorphism group. Both groups are examples of groups of parabolic characteristic three and we identify them from the structure of the normalizer of…

We present an identification theorem for the groups $F_4(2)$ and $Aut(F_4(2))$ based on the structure of the centralizer of an element of order 3.

We describe a construction of an algebra over the field of order 2 starting from a conjugacy class of 3-transpositions in a group. In particular, we determine which simple Lie algebras arise by this construction. Among other things, this…

In this article, we study $2$-designs with $\lambda=2$ admitting a flag-transitive almost simple automorphism group with socle a finite simple exceptional group of Lie type, and we prove that such a $2$-design does not exist. In conclusion,…