Recent content by Hello Kitty

I'm currently looking to use a nonparametric alternative to the parametric repeated-measures with a between-subjects variable. I'm aiming to investigate three sets of subscores of a test taken by three groups of participants. I was initially using the parametric repeated-measures ANOVA using the...

You mean to say that the sufficient condition I proposed is false, not that the original claim is false. (If our constructible set is closed then that set is dense and open in itself.) So any suggestion on how to prove the original claim?

I recently came across the following remark in a book: "Notice that a constructible set contains a dense open subset of its closure." Now this doesn't seem at all obvious to me. Let us recall the definitions first. A locally closed set is the intersection of a closed and an open set. A...

Is it true that the characteristic polynomial of an n by n matrix over GF(q) splits into linear factors over GF(q^n)? I see that it must do if the polynomial is irreducible but what if it isn't?

I wonder why there's not much interest. Too hard, or boring maybe? Possibly I confused people with my erroneous statement: Anyway, just in case anyone is interested, I think I've solved it. It's easier than I thought: Since the roots of f_i lie in \mathbb{F}_{q^{d_i}} and so have order...

Many thanks. Yes, silly of me not to spot this sooner. Actually I'm trying to solve a more general problem and this would have been a sufficient condition if it were true. My post in the algebra section has the details.

Let f = X^n + a_{n-1}X^{n-1} + \cdots + a_1X + a_o be a polynomial over \mathbb{F}_q for some prime power q such that the least common multiple of the (multiplicative) orders of its roots (in \mathbb{F}_{q^n}) is q^n -1. I would like to show that one of these roots has order q^n -1. (I.e. the...

I'm trying to prove or disprove the following: Let a_1, ..., a_n be natural numbers such that the least common multiple of EVERY n-1 of them is equal to lcm(a_1, ..., a_n) = m. Is it true that a_i = m for some i? The method I've tried so far is to build systems of equations using the...

Let H, K be finite groups. Instead of asking what groups G there are such that K can be embedded as a normal subroup and G/K is isomorphic with H (the usual extension problem for groups), I've been thinking about the following: Which groups G exist such that H and K can be embedded as (not...

So the character degrees over \overline{\mathbb F_p} are the same as for \mathbb C provided p \not\vert \ |G| ?

I'm done a basic course on representation theory and character theory of finite groups, mainly over a complex field. When the order of the group divides the characteristic of the field clearly things are very different. What I'd like to learn about is what happens when the field is not...

OK thanks guys. So F(a) has cardinality the square of F in Hurkyl's question, hence a splitting field of an (irreducible) quadratic over F_{p^n} is always F_{p^{2n}}. So we just need the polynomial X^2 + \mu X + \lambda to have a root which is a primitive element of F_{p^{2n}}. I guess...

The q's should be p's. Corrected version: I'm trying to prove that GL(2,p^n) has a cyclic subgroup of order p^{2n} - 1. This should be generated by \left( \begin{array}{cc} 0 & 1 \\ -\lambda & -\mu \end{array} \right) where X^2 + \mu X + \lambda is a polynomial over F_{p^n} such...

I'm trying to prove that GL(2,p^n) has a cyclic subgroup of order p^{2n} - 1. This should be generated by \left( \begin{array}{cc} 0 & 1 \\ -\lambda & -\mu \end{array} \right) where X^2 + \mu X + \lambda is a polynomial over F_{p^n} such that one of its roots has multiplicative order...

What is the definition of an irreducible group. The context is that I have some theorem talking about "an irreducible cyclic subgroup of GL(n,F)". Is it one that can't be written as a product of two other subgroups in a non-trivial way?