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#2: Using trivial Sylow theory, post the list of possibilities and we'll help you from there.
#3: Prove that irreducibles are primes and then show that every element can be written as a product of irreducibles and units. What do you think irreducible elements look like in this ring? Prove it.
One important fact about the Orthogonal group is that it's generated by reflections. Assuming you're ok with this, your goal is to prove that all reflections are in N. Use your previous post to do this.
No, it's all numbers such that they satisfy the property 2 < x < 9. The left of the '|' is the place where your drawling your elements from (in this case, it's all numbers). The right of the '|' is the property that those elements must satisfy. For example:
This is all numbers x in the set [0, infinity) such that 1/x is greater than 1.
{x | 2 < x < 9}
When you read this out loud, it goes:
x is such that 2 is less than x and x is less than 9.
That's why we call '|' that symbol "such that".
When ever checking to see if an ideal is maximal or prime in a commutative ring with 1 (like Z/10Z), the best method tends to be modding out. If you mod out by an ideal and get a field, then the ideal is maximal. If you mod out and get an integral domain, then it's prime.
Use the isomorphism theorem when you're doing this.
Edit: And it should take no longer than 5 minutes to find every ideal of Z/10Z.
Cramer's method
Do you perhaps mean Cramer's rule? It is typically easier to solve the system using row reduction than finding the inverse matrix.
Are you having problems using row reduction? Or do you not know it?
I know that, because N is a normal subgroup, ANA-¹ = N for all A ∈ O(2)
What does this mean, with respect to the reflection that is in N?
I'm not sure how to go about doing this?
There is no way (that I know of) to just look at the group and say, "Ok, here is what the elements are going to be." Remember G has only 120 elements, which really isn't all that many. The task at hand is to reduce this down to a more feasible number of candidates.
The trick to doing this is to assume that the elements exist, and then figure out what properties they would satisfy. So, what properties must these elements satisfy, and how do those properties eliminate elements from G?
Whoops, just be careful with the case where a = d or b = c (or both!) of course.
Am I right in saying that a, b, c and d can only have the values 0, 1, 2, 3 or 4?
Yes.
we wouldn't calculate the determinant modulo 5 would we?
Yes, you would. In general you can consider SL(n, R), where R is any ring. The determinate here is the same as the determinant in the real case, but the elements add and multiply up to 1 in R.
First note that a and d may be switched without affecting the determinant, and the same with b and c. So we may assume a < d and b < c, and then factor in the extra ways later. This trick is an extremely important concept in group theory: Symmetries make calculations easier, but only when you use them.
if infinity is a prime number, then naturally p*T --> 2pi
What is p? What does it mean for infinity to be a prime number?
so that the part intercepted by the coordinate axes is a minimum.
My interpretation of this is:
Let T be line tangent to the ellipse. Set a to be the x-intercept and b to be the y-intercept of T. Now set d to be the distance between a and b, or infinity if one of the intercepts does not exist.
Minimize d.
Is that right?
I.....don't know...
Induction is a very powerful tool when it comes to recurrence because of how much it allows you to assume. Start off with the base case and inductive hypothesis, then do as much as you can till you get stuck.
I cleaned up your latex, and deleted the two subsequent posts. Remember in latex, every symbol (including varphi) requires a '\' before it. But this was not the only issue. Even with proper latex code, it still would not work. I think you copied and pasted the code from somewhere, and formatting (invisible) characters were included in the mix. This confuses and upsets latex.
As to your question: Which method should one use to prove almost any statement about a recurrence relation?
Same method proves mine cannot exist either.
Perhaps, but there is no reason to do all that work again. You've in fact just proven that your polynomial doesn't exist in post #3. Because if the polynomial in post #1 did exist, then the polynomial in post #3 would have to exist (vertical and horizontal shift). That's a contradiction right there.
What part are you having problems on?
So you need to show it's injective, surjective, and a homomorphism. Which part are you having problems with?
Not a python programmer here, but it looks like day is declared locally instead of globally. This means that it won't exist in other local routines.
1. For maximality, typically the trick is to mod out by the ideal then prove you get a field. Remember this works equally well for ideals in quotients of a polynomial ring because of the third isomorphism theorem.
If you have a suspicion that your ring is local, remember to just look at the set of all nonunits.
For rings, it seems that proving something is primary is always done by this proposition:
2. Don't worry about generators for J^n, they exist but they aren't important. Remember what it means to be a generating set: every element is expressible as a "linear combination" with coefficients from R. It's obvious that you want to take your n as something like the highest n from the definition of rad(I). But this won't work. Why? (And think pigeon hole principle)
Is there a certain part of the proof you're having trouble with?
All big-O means that it is an upper bound. big-theta means that it is both an upper and lower bound. It should be obvious from looking at 2^n + 6n^2 + 3n that it is big-theta of 2^n. Show that there exists a k and K such that
k2^n <= 2^n + 6n^2 + 3n <= K2^n
For all n >= c for some c.
For your section one: binomial theorem.
Which part of the problem are you having trouble with?
Algebraic geometry can be summed up (and perhaps wrongly) as the study of polynomials over a field with n variables. We denote
Along those lines, I found this rather fun statement:
Anyone want to attempt a proof?
What grade are you and what math class are you currently in?