Which elements of z42 are invertibles?

  • Thread starter duki
  • Start date
  • Tags
    Elements
In summary, the invertibles in z42 are 1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, and 41. This pattern holds true for all groups, and an interesting fact is that 1 is its own inverse. There is at least one invertible without a partner, meaning it must be its own inverse. It is not always easy to spot, but you can use Fermat's little theorem to determine if a number is relatively prime and thus an invertible in a group.
  • #1
duki
264
0

Homework Statement



which elements of z42 are invertibles?

Homework Equations





The Attempt at a Solution



In my notes I have that invertibles are relatively prime to the order of the group. So I have

1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41.

Is that the case for all groups?
 
Physics news on Phys.org
  • #2


duki said:

Homework Statement



which elements of z42 are invertibles?

Homework Equations





The Attempt at a Solution



In my notes I have that invertibles are relatively prime to the order of the group. So I have

1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41.

Is that the case for all groups?

Is what the case for all groups?

Interesting fact: Note that 1 is its own inverse. That leaves 11 invertibles, so if you repeatedly remove pairs of invertibles, you have at least one left without a partner, meaning it has to be its own inverse. Which one is it? Is there an easy way to spot it? Is there more than one?
 
  • #3


Is what the case for all groups?
Can you always take the relatively primes and get the invertibles?
 
  • #4


duki said:
Can you always take the relatively primes and get the invertibles?

Yes, but do you know why? Hint: can you apply Fermat's little theorem?
 

1. What are invertible elements in z42?

Invertible elements in z42 are those elements that have a multiplicative inverse, meaning that when multiplied by another element, they result in the identity element of z42, which is 1.

2. How do you determine if an element in z42 is invertible?

To determine if an element in z42 is invertible, you can use the Euclidean algorithm to find the greatest common divisor (GCD) between the element and 42. If the GCD is 1, then the element is invertible.

3. Are all elements in z42 invertible?

No, not all elements in z42 are invertible. Only those elements that have a multiplicative inverse are considered to be invertible. For example, 0 is not invertible in z42 because there is no number that can be multiplied by 0 to result in 1.

4. How many invertible elements are there in z42?

There are 12 invertible elements in z42. These are the numbers that are relatively prime to 42, meaning they have a GCD of 1 with 42.

5. What is the notation used for invertible elements in z42?

The notation used for invertible elements in z42 is Z42*, which indicates the set of all invertible elements in z42.

Similar threads

Determine whether the integer 701 is prime by testing?
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Prove there does not exist invertible matrix C satisfying A = CB
    • Calculus and Beyond Homework Help
Replies
6
Views
881
Set theory, intersection of two sets
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Determine the set of odd primes ## p ##?
    • Precalculus Mathematics Homework Help
Replies
14
Views
1K
Group Theory: Finite Abelian Groups - An element of order
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Principle Ideals of a Polynomial Quotient Ring
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Aut(D_12) Elements & Inner Automorphisms: Dihedral Group with 12 Elements
    • Calculus and Beyond Homework Help
Replies
11
Views
1K
QR factorization for a 4x4 tridiagonal symmetric matrix
    • Calculus and Beyond Homework Help
Replies
4
Views
2K
Determining elements in Aut(D_8)
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
MHB Interesting property about 2019( the current year)
    • General Discussion
Replies
4
Views
2K

Hot Threads

Recent Insights

Back
Top