Proving Lagrange's Theorem I had no idea how to start the proof so I cheated and looked it up, and the proof that I can understand uses cosets. How do you know that you should start with cosets to perform this proof? I spent about half the exam time trying to think of ideas and ways to try and prove that the order of an element divides the order of a group, but ultimately I ran out of time and had to leave the problem blank. I would never have thought to use cosets, and didn't even know that cosets of a subgroup partitioned the group, much less be able to prove that on the spot during a test. So I guess my question is, how do you know what concepts will work for a proof, considering you have so much knowledge stored of the subject from class? Out of all of that, you have to specifically choose one particular concept and derive the proof using that, but how do you gain the insight to do this? Also, is there an easier approach to proving Lagrange's theorem more directly instead of trying to come up with something clever like using coset partitions?
 A: One of the most recurring themes used while counting some finite set is using an equivalence relation on the set.It gives a partition of the set into disjoint classes. Now when one defines cosets using some subgroup $H$ of a group $G$, then one is basically defining an equivalence relation on $G$ thus dividing into disjoint equivalence classes, one of which is $H$ itself. One more advantage of cosets is that it gurantees that cardianlity of each equivalence class is same and is equal to $|H|$, thus giving a very powerful tool for dividing $G$ in portions each of which have size $|H|$.
Edit : - Given a group $G$ and its subgroup $H$ one defines $\forall g_1,g_2\in G$, $g_1\sim g_2$ iff $g_2^{-1}g_1\in H$. this is an equivalence relation on $G$ and you'll see that the equivalence classes are precisely the cosets of $H$ in $G$. Well it can be made intuitive. Suppose you want to count the number of objects in a box. If you can't count it one by one then one way is to divide the objects in the box into seperate pieces and then count each piece seperately. That is the idea of equivalence classes. With cosets, those seperate pieces have same number of objects. For examples suppose your box has name $G$ and it has $20$ balls. Divide it into $4$ seperate boxes the first one named $H$. Suppose you know beforehand that all of your boxes contain equal number of balls as $H$, then its pretty clear that each will have $5$ balls, and clearly $5|20$.
A: Proving on demand under time pressure is difficult. It just is. It isn't a realistic reflection of mathematics research, and I hope you are cutting yourself some slack.
I can't speak to your specific question because when I first learned group theory, the relevant section of the book was called "Cosets and the Theorem of Lagrange." So, hard to miss. I'm a little surprised this was open as an exam question, since that would suggest it wasn't covered directly, which I would not have expected. 
