Recent content by Jösus

Correct me if I'm wrong, but shouldn't the equation read dg/dt = f(g(t)), \quad dh/dt = f(h(t)), and thus there would be no apparent reason for these derivatives to be equal? I have thought about it some more, and found that if f(x_{0}) \neq 0 then there is an interval containing x_{0} on...

Hello! I would like to prove the following statement: Assume f\in C^{1}(\mathbb{R}). Then the initial value problem \dot{x} = f(x),\quad x(0) = x_{0} has a unique solution, on any interval on which a solution may be defined. I haven't been able to come up with a proof myself, but would...

Hi! I'm studying for an exam in group- and ring theory, and I have some questions about two problems that I have not managed to solve. I would greatly appreciate help. Problem 1. Determine the order of the group G with the presentation (a,b \big\vert\: a^{6} = 1, b^{2} = a^{3}, ba = a^{-1}b)...

Hello! I'm currently taking a course in group- and ring theory, and we are now dealing with a chapter on finitely generated modules over PIDs. I have stumbled across some problems that I can't really get my head around. It is one in particular that I would very much like to understand, and I...

If "A Classical Introduction to Modern Number Theory" would happen not to fit, I recommend reading "An Introduction to Number Theory", by G. Everest and T. Ward. As with the first, it is on the whole not an undergraduate-level text, but the first few chapters are not too complicated, and would...

First of all; thank you for taking the time to read and post. However, when thinking about this problem a bit more, it seems like my problems goes deeper than I first though. I simply haven't understood the concept well enough yet, or at least, I still find it a bit too technical to be able to...

Hello Lately, I have been studying some group theory. On my own, I should add, so I don't really have any professor (or other knowledgeable person for that matter) to ask when a problem arises; which is why I am here. I had set out to find all small groups (up to order 30 or something), up to...

I came up with something that seems to be proving the assertion. Though, I would not say that I am sure that everything in it is correct. Could someone please read through and comment on it? Here it goes: First I define a concept that I use in the proof. I have no idea if what I define is...

What about: Assume on the contrary that there does not exist any k \in \mathbb{Z}^{+} such that, for g \in G, g^{k} = 1_{G}, the identity. As G is finite, we can't have all g^{k} distinct for all positive integers k, as it would result in a set having infinite cardinality. Thus \exists...

I am sorry for that mistake. Figured it out when thinking of subgroups generated by a double transposition like (1,2)(2,3). It is indeed silly when thinking about it. However, I would like to avoid using Cauchy's Theorem, as it is presented as an excercise far earlier than the theorem in the...

I'm sorry, but I can't really make sense out of your proof. Not because I think it is incorrect, but I can't understand your notation. What is, for example M? If I understand your notation correctly, you introduce an equivalence relation to show the existence of a least positive integer k with...

I'm not sure I understand your formulation. Or, it is your partial solution I do not grasp. Do I interpret i correctly if I'd restate it as follows? Let a be a k-cycle in Sn. Show that if a = t1 * t2 * ... * ts, where the ti are transpositions for 1 = 1, 2, ..., s, then s must be greater than...

I guess if f: G --> G' is an isomorphism of abelian groups, H is a subgroup of G it follows directly (from some correspondence thm for groups) that f(H) is a subgroup of G'. As they are abelian and thus normal, we can consider the quotient group G'/f(H). Define now the map h: G/H --> G'/f(H) by...

Hello, I am quite new here, as my number of posts might indicate. Thus I am not really sure whether or not this question should be posted here or somewhere else. It is not a homework, but neither is it a question that could not be a homework. However, here we go. I have, during a course in...

Homework Statement I am trying to prove that the alternating group on five letters, A5, contains no subgroups of order 20. Homework Equations I guess nothing is needed here, for this problem. Though I will use this extra space to explain my notation, if it would happen to differ from the...