Recent content by the_fox

This is a problem I encountered in Martin Isaacs' 'Finite Group Theory'. It's located at the end of Chapter II which deals with subnormality, and the particular paragraph is concerned with a couple of not so well-known results which I quote for reference: (In what follows F is the Fitting...

That's how X is defined. What exactly do you mean?

Let G be a group and A a subset of G with n elements such that if x is in A then x^(-1) is not in A. Let X={(a,b) a in A, b in A, ab in A}. Prove that X contains at most n(n-1)/2 elements.

But, how do we know that every singer cycle in GL(n,q) corresponds to a linear transformation T=ax, where a is a primitive element?

So how can we get commutativeness into the picture?

A nice example indeed. The part I didn't really get though, was about the action. If an element commutes with A in this example, then what? I assume your selection was motivated by taking the companion matrix of the 2-degree primitive polynomial over GF(2).

I'm not even sure I understand what you ask. Can you be a little bit more analytic, or perhaps illustrate this with an example?

Oops.. sorry for the double post.

I've also noticed that s.c. commute only with their powers, i.e., are self centralising. Can we prove that?

I just realized that S. cycles are also self centralizing, but I can't prove it. If I manage to though, I will have a formula for how many there are.

What I mean is, that Singer cycles with the same minimal polynomial belong to the same conjugacy class defined by C(m(x)), the companion matrix of m(x). Assume conjugacy is restricted to the group theoretic sense. For example, all Singer cycles in GL(3,2) belong to either one of the two distinct...

By the way, I think it's true that no element in GL(n,q) has order that exceeds q^n-1. An thoughts on this?

I think it becomes obvious if you notice that a Singer cycle cannot have an irreducible polynomial that is not primitive as a minimal polynomial. For example, can f(x)=x^4 + x^3 + x^2 +x + 1 in F_16[x] be the minimal polynomial of a singer cycle in GL(4,2)?

I'm sorry; I meant φ(q^n-1)/n classes. the companion matrices that correspond to primitive polynomials of degree n have order q^n-1 and Singer cycles that have the same minimal polynomial necessarily lie in the same conjugacy class.

Allright. So we can say that elements of order q^n-1 in GL(n,q) can be divided in φ(q^n-1) conjugacy classes, to combine results with the Singer cycles thread. What I want to do is find the order of each conjugacy class. Any ideas?