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Which elements of z42 are invertibles?

  • Thread starter duki
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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?
 

Answers and Replies

  • #2
jbunniii
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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?
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?
 
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Is what the case for all groups?
Can you always take the relatively primes and get the invertibles?
 
  • #4
jbunniii
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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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