Is U(p^k) Cyclic and How Does It Relate to Modular Arithmetic?

  • Thread starter Thread starter Dead Metheny
  • Start date Start date
  • Tags Tags
    Cyclic
Click For Summary
SUMMARY

The discussion centers on the cyclic nature of the group U(p^k), where p^k is an odd prime power. It is established that U(p^k) is cyclic, with the order of the group being p^k - p^(k-1). The hint provided by the professor regarding the solutions to x^2 = 1 mod p^k is crucial for understanding the structure of the group. Additionally, it is noted that U(2n) is not cyclic for n > 2 due to the presence of two elements of order 2.

PREREQUISITES
  • Understanding of group theory, specifically cyclic groups
  • Familiarity with modular arithmetic and its properties
  • Knowledge of the structure of U(n) groups
  • Basic concepts of prime powers and their implications in group theory
NEXT STEPS
  • Study the properties of cyclic groups in group theory
  • Learn about the structure and properties of U(n) groups
  • Explore modular arithmetic techniques, particularly focusing on solving equations like x^2 = 1 mod n
  • Investigate the implications of prime powers in group structures
USEFUL FOR

This discussion is beneficial for mathematicians, particularly those specializing in abstract algebra, students studying group theory, and anyone interested in the properties of modular arithmetic and cyclic groups.

Dead Metheny
Messages
3
Reaction score
0
Prove that U(p^k) is cyclic

p^k is an odd prime power.

I've been working on this problem for a while and can't figure it out. The professor's hint is "to think about the solutions to x2 =1." (pk - 1)2 mod pk = 1 but I'm unsure how that is helpful.

I know that that 2 generates every set by trial and error, and I'm reasonable sure that any prime less than p generates the set. The order of the set is pk - pk - 1.

If |q| = n, then q|U(pk)| = 1 mod pk and n | p - 1 or n | pk - 1 or n | (p - 1)pk - 1. I'm unsure of how to solve the modular arithmetic from here though.

Edit: I wrote down wrong question for title, apologies. U(2n) is not cyclic for n>2 because it will contain two elements of order 2.
 
Last edited:
Physics news on Phys.org
I'm still stuck on this. Anyone able to lend a hand?
 

Thread 'Distance between a Clock's hands when the distance is increasing most rapidly'

Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?
View full post »

Similar threads

Prove that ## 1^{p}+2^{p}+3^{p}+\dotsb +(p-1)^{p}\equiv 0\pmod {p} ##.
  • · Replies 6 ·
Replies
6
Views
3K
Congruence and Primes: Proving the Order of an Integer Modulo a Prime
  • · Replies 4 ·
Replies
4
Views
1K
Undergrad How Do You Calculate Traces in Compton Scattering Feynman Amplitudes?
  • · Replies 15 ·
Replies
15
Views
3K
If ## p\neq5 ## is an odd prime, prove that either ## p^{2}-1 ## or....
  • · Replies 15 ·
Replies
15
Views
4K
Prove ##|\{ q\in\mathbb{Q}: q>0 \} |=|\mathbb{N}|##.
  • · Replies 3 ·
Replies
3
Views
2K
Conditional expectations related to count of event occurring kth time
  • · Replies 2 ·
Replies
2
Views
1K
Modular arithmetic on vector spaces
  • · Replies 2 ·
Replies
2
Views
1K
Groups that cannot be the direct product of subgroups
  • · Replies 1 ·
Replies
1
Views
2K
Can Discrete Logarithms Be Added in Modular Arithmetic?
  • · Replies 16 ·
Replies
16
Views
2K
Showing that ##Aut ~ Z_p \simeq Z_{p-1}##
  • · Replies 10 ·
Replies
10
Views
3K