Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the p-width of finite simple groups
134
0
0.0
(
0
)
Authors
Alexander J. Malcolm
Ask ChatGPT about the research
No Arabic abstract
In this paper we measure how efficiently a finite simple group $G$ is generated by its elements of order $p$, where $p$ is a fixed prime. This measure, known as the $p$-width of $G$, is the minimal $kin mathbb{N}$ such that any $gin G$ can be written as a product of at most $k$ elements of order $p$. Using primarily character theoretic methods, we sharply bound the $p$-width of some low rank families of Lie type groups, as well as the simple alternating and sporadic groups.
rate research
Read More
We give a short proof of the fact that if all characteristic p simple modules of the finite group G have dimension less than p, then G has a normal Sylow p-subgroup.
For a finite group generated by involutions, the involution width is defined to be the minimal $kinmathbb{N}$ such that any group element can be written as a product of at most $k$ involutions. We show that the involution width of every non-abelian finite simple group is at most $4$. This result is sharp, as there are families with involution width precisely 4.
We define and study supercharacters of the classical finite unipotent groups of symplectic and orthogonal types (over any finite field of odd characteristic). We show how supercharacters for groups of those types can be obtained by restricting the supercharacter theory of the finite unitriangular group, and prove that supercharacters are orthogonal and provide a partition of the set of all irreducible characters. We also describe all irreducible characters of maximum degree in terms of the root system, and show how they can be obtained as constituents of particular supercharacters.
Let $p$ be a fixed prime. For a finite group generated by elements of order $p$, the $p$-width is defined to be the minimal $kinmathbb{N}$ such that any group element can be written as a product of at most $k$ elements of order $p$. Let $A_{n}$ denote the alternating group of even permutations on $n$ letters. We show that the $p$-width of $A_{n}$ $(ngeq p)$ is at most $3$. This result is sharp, as there are families of alternating groups with $p$-width precisely 3, for each prime $p$.
This is a nearly complete manuscript left behind by Boris Weisfeiler before his disappearance during a hiking trip in Chile in 1985. It is posted on a request from the authors sister, Olga Weisfeiler.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
