4
votes
1answer
163 views

orthogonal group in characteristic 2

Let $O(2,\mathbb{Z}_2)$ be the orthogonal group of order two matrices. On $\mathbb{Z}_2$ there should exist just one odd quadratic form, hence the stabilizer subgroup $O^-$ of an odd quadratic for ...
0
votes
0answers
71 views

Eigenvalues of the Cayley-like graph

Let $ F_q $ be a finite field of characteristic 2. Let $ x^2 + Sx +P \in F_q[x] $ be an irreducible polynomial over $ F_q $, and let $ g $ be one of its roots in $ F_{q^2} $. Define a map $ M: ...
7
votes
0answers
176 views

A strong sum-product “for translates” in finite fields

In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting? Proposition There exists an absolute constant $c$ such ...
4
votes
1answer
170 views

Cyclic subgroups of GL(n,q)

Let $q$ be a prime power. It is well known that every Singer subgroup (subgroup of order $q^n-1$) in $GL(n,q)$ are conjugate. My question is: If $H$ is a cyclic subgroup of order $m$ in $GL(n,q)$, ...
18
votes
8answers
2k views

Fantastic properties of Z/2Z

Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ...
1
vote
2answers
260 views

When is PSU(2,q^2) = PSL(2,q) ?

The context for this question is from page 284 - 287 of Berger's paper: ...
7
votes
4answers
1k views

Orthogonal Groups over finite fields

Hello Let $\mathbb{F}_p$ be the finite field with $p$ elements. One can show that over finite fields, there are just two non-degenerate quadratic forms. So here I want to pick any non-degenerate ...
5
votes
1answer
282 views

Aschbacher classes and $\mathbb{F}_p$-subspace stabilizers in classical linear groups

I am reading the Kleidman-Liebeck book ("The subgroup structure of the finite classical groups") which is about the Aschbacher classification of maximal subgroups of the classical almost simple ...
10
votes
3answers
937 views

Can we always find such an irreducible polynomial of degree n where degree(p(x)-x^n)<= n/2?

Consider unitary polynomials of degree $n$ over $GF(2)$. That is, polynomials of the form $p(x) = \sum_{i=0}^n a_i x^i$ where $a_i\in GF(2)$ and $a_n=1$. Can we always find such an irreducible ...
5
votes
7answers
540 views

Classifications of finite simple objects

I'm curious to know if other classifications are known of "finite simple structures" in the same spirit of the monumental classification of finite simple groups. Here I mean "classification" in the ...
6
votes
1answer
1k views

Order of finite unitary group

This may be an easy exercise but I am not getting it. Let $\mathbf F_q$ be a finite field with $q$ elements and $\mathbf F_{q^2}$ be its degree two extension. Define an automorphism $\sigma$ of ...
17
votes
3answers
877 views

Non-split extensions of $GL_n(F_q)$ by $F_q^n$ ?

A very naive question : I just learned that there is a non-split extension of $GL_3(F_2)$ by $F_2^3$ (with standard action). It can be realized as the subgroup of the automorphism group $G_2$ of ...
3
votes
2answers
939 views

Cyclic order relation in Zn

The ring Zn:={0,1,..,n-1} under addition and multiplication modulo n. Suppose a,b,c,x $\in$ Zn are nonzero and the cyclic order R(a,b,c) holds, then under what conditions does R(ax,bx,cx) hold ?
23
votes
9answers
2k views

How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?

Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings, fields, graphs, partial orders, etc. ...