Let G be a finite group and $\mathrm {Irr}(G)$ the set of all irreducible complex characters of G. Define the codegree of $\chi \in \mathrm {Irr}(G)$ as $\mathrm {cod}(\chi ):={|G:\mathrm {ker}(\chi ) |}/{\chi (1)}$ and denote by $\mathrm {cod}(G):=\{\mathrm {cod}(\chi ) \mid \chi \in \mathrm {Irr}(G)\}$ the codegree set of G. Let H be one of the $26$ sporadic simple groups. We show that H is determined up to isomorphism by cod$(H)$.

Let $G$ be a finite permutation group of degree $n$ and let ${\rm ifix}(G)$ be the involution fixity of $G$, which is the maximum number of fixed points of an involution. In this paper, we study the involution fixity of almost simple primitive groups whose socle $T$ is an alternating or sporadic group; our main result classifies the groups of this form with ${\rm ifix}(T) \leqslant n^{4/9}$. This builds on earlier work of Burness and Thomas, who studied the case where $T$ is an exceptional group of Lie type, and it strengthens the bound ${\rm ifix}(T) > n^{1/6}$ (with prescribed exceptions), which was proved by Liebeck and Shalev in 2015. A similar result for classical groups will be established in a sequel.

The first author [J. Brough, ‘On vanishing criteria that control finite group structure’, J. Algebra458 (2016), 207–215] has shown that for certain arithmetical results on conjugacy class sizes it is enough to consider only the vanishing conjugacy class sizes. In this paper we further weaken the conditions to consider only vanishing elements of prime power order.

After summarizing from previous papers the definitions of the concepts associated with nets, i.e. triples of 6-transpositions in the Monster up to braiding, we give some results.

In this paper we determine the suborbits of Janko’s largest simple group in its conjugation action on each of its two conjugacy classes of involutions. We also provide matrix representatives of these suborbits in an accompanying computer file. These representatives are used to investigate a commuting involution graph for J4.

Supplementary materials are available with this article.

This paper establishes recognition theorems for the finite Lie-type groups defined in odd characteristic $p$. The hypotheses of these theorems are couched in terms of certain local data in the form of an amalgam which is called a weak $BN$-pair of rank 2. The groups to be recognized are completions of these amalgams which satisfy the condition of being of local characteristic $p$ (a condition satisfied by the finite Lie-type groups defined in characteristic $p$). The results presented will find application in one of the ongoing revisions of the finite simple group classification.

The alternating group of degree 6 is located at the junction of three series of simple non-commutative groups: simple sporadic groups, alternating groups and simple groups of Lie type. It plays a very special role in the theory of finite groups. We shall study its new roles both in a finite geometry of a certain pentagon in the Leech lattice and also in the complex algebraic geometry of K3 surfaces.

The structure of the point-line collinearity graph of the maximal 2-local geometry for the Monster simple group is investigated. The main results describe the first two discs around an arbitrary vertex.

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.

A locally projective amalgam is formed by the stabilizer $G(x)$ of a vertex $x$ and the global stabilizer $G\{x,y\}$ of an edge containing $x$ in a group $G$, acting faithfully and locally finitely on a connected graph $\Gm$ of valency $2^n\,{-}\,1$ so that (i) the action is 2-arc-transitive, (ii) the sub-constituent $G(x)^{\Gm(x)}$ is the linear group ${\rm SL}_n(2) \,{\cong}\, {\rm L}_n(2)$ in its natural doubly transitive action, and (iii) $[t,G\{x,y\}] \,{\le}\, O_2(G(x) \,{\cap}\, G\{x,y\})$ for some $t \,{\in}\, G\{x,y\} \setminus G(x)$. Djoković and Miller used the classical Tutte theorem to show that there are seven locally projective amalgams for $n\,{=}\,2$. Trofimov's theorem was used by the first author and Shpectorov to extend the classification to the case $n \,{\ge}\, 3$. It turned out that for $n\,{\geq}\,3$, besides two infinite series of locally projective amalgams (embedded into the groups ${\rm AGL}_n(2)$ and $O_{2n}^+(2)$), there are exactly twelve exceptional ones. Some of the exceptional amalgams are embedded into sporadic simple groups $M_{22}$, $M_{23}$, $Co_2$, $J_4$ and $\hbox{\it BM}$. For a locally projective amalgam $\cA$, the minimal degree $m\,{=}\,m(\cA)$ of its complex representation (which is a faithful completion into ${\rm GL}_m(\CC)$) is calculated. The minimal representations are analysed and three open questions on exceptional locally projective amalgams are answered. It is shown that

$\cA_4^{(1)}$ possesses ${\rm SL}_{20}(13)$ as a faithful completion in which the third geometric subgroup is improper;

$\cA_4^{(2)}$ possesses the alternating group ${\rm Alt}_{64}$ as a completion constrained at levels 2 and 3;

$\cA_4^{(5)}$ possesses ${\rm Alt}_{256}$ as a completion which is constrained at level 2 but not at level 3.

A $5$-local characterization of the Lyons sporadic simple group is presented.

A locally projective amalgam is formed by the stabilizer G(x) of a vertex x and the global stabilizer G{x, y} of an edge (containing x) in a group G, acting faithfully and locally finitely on a connected graph Γ of valency 2n - 1 so that (i) the action is 2-arc-transitive; (ii) the subconstituent G(x)Γ(x) is the linear group SLn(2) = Ln(2) in its natural doubly transitive action and (iii) [t, G{x, y}] < O2(G(x) n G{x, y}) for some t G G{x, y} \ G(x). D. Z. Djokovic and G. L. Miller [DM80], used the classical Tutte’s theorem [Tu47], to show that there are seven locally projective amalgams for n = 2. Here we use the most difficult and interesting case of Trofimov’s theorem [Tr01] to extend the classification to the case n > 3. We show that besides two infinite series of locally projective amalgams (embedded into the groups AGLn(2) and O2n+(2)) there are exactly twelve exceptional ones. Some of the exceptional amalgams are embedded into sporadic simple groups M22, M23, Co2, J4 and BM. For each of the exceptional amalgam n = 3, 4 or 5.

A construction of the Monster simple group is described implicitly as $196882\,{\times}\,196882$ matrices over the field of 3 elements.