Which elements of z42 are invertibles?

[duki]

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?

 

[jbunniii]

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?

 

[duki]

Is what the case for all groups?

Can you always take the relatively primes and get the invertibles?

 

[jbunniii]

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?