Linked Questions

197
votes
91answers
54k views

Which one result in mathematics has surprised you the most? [closed]

A large part of my fascination in mathematics is because of some very surprising results that I have seen there. I remember one I found very hard to swallow when I first encountered it, was what is ...
70
votes
10answers
34k views

Examples of finite nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
32
votes
2answers
2k views

Rubik's Cube Not a Group?

I read online that although the 3x3x3 is a great example of a mathematical group, larger cubes aren't groups at all. How can that be true? There is obviously an identity and it is closed, so that ...
24
votes
1answer
905 views

If $S$ is a nonempty subset of group $G$, then $S^{|G|}$ is a subgroup of $G$.

Let $G$ be a group with $|G| = n$ and let $ \emptyset \ne S \subseteq G$. I want to show that $S^n$ is a subgroup of $G$ where by $S^n$ I mean the set $\lbrace s_1\cdots s_n \; | \; s_i \in S\rbrace$...
15
votes
2answers
294 views

Examples of sets which are not obviously sets

In my (limited) experience, it is usually easy to see when something is large enough to be a proper class, by constructing an element of the class for every set. However, sometimes such a proper ...
6
votes
1answer
1k views

Nice examples of finite things which are not obviously finite

This question is in the spirit of the question "Nice examples of groups which are not obviously groups". There are many impressive finiteness results in mathematics. For example: The finiteness of $\...
8
votes
1answer
509 views

In what way is combinatorial game theory connected to the rest of mathematics?

Since my University library lists Conway's "Winning ways for your mathematical plays in the section "recreational mathematics" alongside books on origami and puzzles, I wondered to what extent game ...
3
votes
1answer
253 views

Soft question about irreducible representations

So I'm a beginner in representation theory and I'm a bit confused about the 'point' of irreducible representations. From what I understand, irreducible representations are important because they allow ...
4
votes
4answers
61 views

Examples of application of theories to completely unexpected structures?

AFAIK, every mathematical theory (by which I mean e.g. the theory of groups, topologies, or vector spaces), started out (historically speaking) by formulating a set of axioms that generalize a ...
2
votes
1answer
71 views

Set equipped with some operation which is such that it is not known is it a group and is of some importance in mathematics

Well, I know of some examples of groups which are trivial enough and of some which maybe are not so trivial. It could be the case that we could construct some operation on some set which is such that ...