I had been looking lately at Sylow subgroups of some specific groups and it got me to wondering about why Sylow subgroups exist. I'm very familiar with the proof of the theorems ( …

Take two vector spaces $V$ and $W$ over a field $F$. One may form the tensor product $V\otimes W$ and it fulfills an universal property. Elements of $V\otimes W$ are called tensors …

I am interested in theorems with unexpected conclusions. I don't mean an unintuitive result (like the existence of a space-filling curve), but rather a result whose conclusion seem …

It's excellent news that the LMS are to re-publish Cassels & Fröhlich. There are many other excellent mathematics books which are just about impossible (or at least very expen …

As a non-native English speaker (and writer) I always had the problem of understanding the distinction between a 'Theorem' and a 'Proposition'. When writing papers, I tend to name …

[Follow-up to Can every finite graph be represented by one prescribed sequence of natural numbers?, reformulated thanks to a hint from Jacques Carette] Let $V(n,\nu)$ and $E(n, …

(This is a follow-up to my previous question Can every finite graph be represented by an arithmetic sequence of natural numbers?) Since it is obviously false that every finite gra …

With the flavour of this question: http://mathoverflow.net/questions/15366/which-journals-publish-expository-work, I would like to ask the following question Which journals p …

Consider sequence $s(n) = \sin{nx}$. Are there values of $x$ for which the following holds: For every $y \in [-1,1]$ there is a subsequence of $s(n)$ converging to $y$? (Or perhaps …

I'm merely a grad student right now, but I don't think an exploration of the sporadic groups is standard fare for graduate algebra, so I'd like to ask the experts on MO. I did a li …

Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of? Standard community wik …

I hope this question isn't too open-ended for MO --- it's not my favorite type of question, but I do think there could be a good answer. I will happily CW the question if commente …

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please) I'd …

In "continuous" mathematics there are several important notions such as covering space, fibre bundle, Morse theory, simplicial complex, differential equation, real numbers, real pr …

In forcing, we take a collection of forcing conditions and impose a partial order on them. The convention is that if $p$ is stronger than $q$, then we say $p < q$. This is perfe …