Which elements of z42 are invertibles?

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SUMMARY

The elements of the group Z42 that are invertible are those that are relatively prime to 42. The specific invertible elements identified are 1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, and 41. This conclusion is based on the property that an element is invertible if it shares no common factors with the order of the group. Additionally, it is noted that 1 is its own inverse, and the discussion raises the question of whether this property applies universally across all groups.

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  • Understanding of group theory and the concept of invertibility
  • Familiarity with the properties of relatively prime numbers
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  • Basic comprehension of Fermat's Little Theorem
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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?
 
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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?
 


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


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?
 

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