McKay graphs for alternating and classical groups

Let G be a finite group, andαa nontrivial character of G. The McKay graph M (G,α) has the irreducible characters of Gas vertices, with an edge fromχ1toχ2ifχ2is a constituent ofαχ1. We study the diameters of McKay graphs for finite simple groups G. For alternating groups G=An, we prove a conjecture made in [20]: there is an absolute constant C such that diam M (G,α)≤ C log | G| log α (1)for all nontrivial irreducible characters α of G. Also for classical groups of symplectic or orthogonal type of rank r, we establish a linear upper bound Cr on the diameters of all nontrivial McKay graphs. Finally, we provide some sufficient conditions for a productχ1χ2···χlof irreducible characters of some finite simple groups G to contain all irreducible characters of G as constituents

Irreducible subgroups of algebraic groups

A closed subgroup of a semisimple algebraic group G is said to be G‐irreducible if it lies in no proper parabolic subgroup of G. We prove a number of results concerning such subgroups. Firstly they have only finitely many overgroups in G; secondly, with some specified exceptions, there exist G‐irreducible subgroups of type A1; and thirdly, we prove an embedding theorem for G‐irreducible subgroup

On the length and depth of finite groups

An unrefinable chain of a finite group is a chain of subgroups = 0> 1>⋯> =1 , where each is a maximal subgroup of −1 . The length (respectively, depth) of is the maximal (respectively, minimal) length of such a chain. We studied the depth of finite simple groups in a previous paper, which included a classification of the simple groups of depth 3. Here, we go much further by determining the finite groups of depth 3 and 4. We also obtain several new results on the lengths of finite groups. For example, we classify the simple groups of length at most 9, which extends earlier work of Janko and Harada from the 1960s, and we use this to describe the structure of arbitrary finite groups of small length. We also present a number‐theoretic result of Heath‐Brown, which implies that there are infinitely many non‐abelian simple groups of length at most 9. Finally, we study the chain difference of (namely the length minus the depth). We obtain results on groups with chain differences 1 and 2, including a complete classification of the simple groups with chain difference 2, extending earlier work of Brewster et al. We also derive a best possible lower bound on the chain ratio (the length divided by the depth) of simple groups, which yields an explicit linear bound on the length of / ( ) in terms of the chain difference of , where ( ) is the soluble radical of

A note on the probability of generating alternating or symmetric groups

We improve on recent estimates for the probability of generating the alternating and symmetric groups $\mathrm{Alt}(n)$ and $\mathrm{Sym}(n)$. In particular we find the sharp lower bound, if the probability is given by a quadratic in $n^{-1}$. This leads to improved bounds on the largest number $h(\mathrm{Alt}(n))$ such that a direct product of $h(\mathrm{Alt}(n))$ copies of $\mathrm{Alt}(n)$ can be generated by two elements

Multiplicity-free representations of algebraic groups II

We continue our work (started in ``Multiplicity-free representations of algebraic groups", arXiv:2101.04476), on the program of classifying triples $(X,Y,V)$, where $X,Y$ are simple algebraic groups over an algebraically closed field of characteristic zero with $X<Y$, and $V$ is an irreducible module for $Y$ such that the restriction $V\downarrow X$ is multiplicity-free. In this paper we handle the case where $X$ is of type $A$, and is irreducibly embedded in $Y$ of type $B,C$ or $D$. It turns out that there are relatively few triples for $X$ of arbitrary rank, but a number of interesting exceptional examples arise for small ranks.Comment: 60 page