If $G$ is a finite group then the composition table of $G$ is a latin square (ie, each row and column contains each group element exactly once).
Assume now that $|G| = n^2$ for some natural number $n$. We can then split the composition table for $G$ into $n^2$ $n\times n$ squares, and we can ask...
@awllower yes, there is a unique such homomorphism, since the domain is cyclic, so it has a unique subgroup of index $2$ (and a non-trivial homomorphism must be surjective)
(but of course, that uses the existence of a primitive root)
BTW, a saying in the book on finite groups by IMIssacs is quite interesting: representations: homomorphisms to GLV old group theory: homo to S_n transfer: homo to subgroups.
I wonder if it is true... http://math.stackexchange.com/questions/309251/is-it-true-that-the-book-calculate-primes-has-found-the-pattern
I suggest that you need to re-calibrate your estimation of difficulty and/or the talents of high school students.
While a question like that is most certainly accessible to some high school students, those students are exceptional.
And a question of that difficulty is well above the average difficultly level of a true high-school level problem. Even though it might not require techniques which are, individually, more complicated than what a typical HS calculus student knows, it is the comprehensive understanding that becomes more difficult
Fundamentally, driving a Formula 1 car is the same combination of actions as driving my SUV; but Monza is not an every day commute.
A high school student that can solve that problem is definitely advanced in her knowledge, and upon attending university will almost certainly start at least 1 or 2 classes ahead of her peers.
@Arkamis: actually this problem is created by my brother and guess what? The requirement to me is to evaluate it by only using high school knowledge and NO PEN AND PAPER. Now, if you say again that I'm a lier then I understand you.
@Chris'ssisterandpals That means nothing. As I said, driving a formula 1 car requires only pressing a gas pedal, shifting, and steering. That doesn't mean it's a car that a high-schooler can drive.
@user58512: hold on! Chris has some thousands of such problems! I'm thinking to make a mega compilation, but this will take a lot of time (and for many problems I don't have solutions yet). I love so much these little wonders.
@PeterTamaroff: the point is to make the guess that $\lim_{n\to\infty}\int_0^1 \frac{dx}{1+x^n}\, \mathrm{d}x=1$. That is the case $1^{\infty}$. Then we may use the limit $\lim_{x\to0} (1+x)^{(1/x)}=e$
If a question I asked got an answer but not to my satisfaction how can I resurrect it withotu reposting? as it seems to be ignored even though I commented that the answer is not satisfactory
I know the initial ordering with the 100 baskets had equally likely arrangements, I'm just wondering why it reflects towards our case that they are also equally likely
Show that $(m^2 - n^2, 2mn, m^2 + n^2)$ is a primitive Pythagorean triplet
First, I showed that $(m^2 - n^2, 2mn, m^2 + n^2)$ is in fact a Pythagorean triplet.
$$\begin{align*} (m^2 - n^2)^2 + (2mn)^2 &= (m^2 + n^2)^2 \\
&= m^4 -2m^2n^2 + n^4 + 4m^2n^2 \\
&= m^4 + 2m^2n^2 + n^4 \\
&...
