Proof of the Order Divisibility Property

squire636
Messages
38
Reaction score
0
The Order Divisibility Property states that if an = 1 (mod p), then the order ep(a) of a (mod p) divides n.

How can I go about proving this?

Additionally, if a is relatively prime to p, when does the congruence am = an (mod p) hold? Is there a proof for this as well?

Thanks!
 
Physics news on Phys.org
Hint for 1: The division algorithm says we can write ##n=qe_p(a)+r##. What can you say about r?

Hint for 2: If a is relatively prime to p, what is ##e_p(a)##?
 
For the first one, r must equal zero, but I'm not sure how to show that it is true.

For the second, would the order ep(a) = p-1 ?
 
squire636 said:
For the second, would the order ep(a) = p-1 ?

Not really; other integer less than p-1 could do the trick. Anyway, I believe morphism was simply trying to make you think about the very definition of 'order'. And what would you need to do to this definition in order to transform it into something like the premises that you are given.
 
As for the first problem, the division algorithm tells us that ##0 \leq r < e_p(a)##. What can you do with this, squire636?
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I How to Prove a Congruence Relation for Prime Numbers?
Replies
17
Views
2K
I Question on Primitive Roots and GCDs
Replies
2
Views
2K
Show that the sum of twin primes ## p ## and ## p+2 ## is divisible?
Replies
17
Views
3K
I Fundamental theorem of arithmetic from Jordan-Hölder theorem
Replies
5
Views
2K
I ##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?
Replies
31
Views
1K
B Inductive proof for multiplicative property of sdet
Replies
1
Views
1K
I Trouble understanding an online solution to an exercise in Dummit & Foote
Replies
15
Views
635
I How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?
Replies
48
Views
3K
I Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'
Replies
3
Views
2K
I Meaning of "passage to the quotient"?
Replies
3
Views
439

Hot Threads

Recent Insights

Back
Top