Proving that 'a' can be written as the nth power

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Homework Statement



Let a belong to a group and |a| = m. If n is relatively prime to m, show that
a can be written as the nth power of some element in the group.

Homework Equations



We know :
|a| = m
gcd(m,n) = 1

We want to show :
bn = a for some b in the group.

The Attempt at a Solution



Okay, I didn't really know where to start with this one, but I'll give it a try.

We know the gcd can be written as a linear combination, that is :

gcd(m,n) = 1 = ms + nt for some integers s and t.

Now :
1 = ms + nt
a1 = ams + nt
a = ams ant
a = easant
a = asant

Here's where I get stuck. Any pointers?
 
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Zondrina said:

Homework Statement



Let a belong to a group and |a| = m. If n is relatively prime to m, show that
a can be written as the nth power of some element in the group.

Homework Equations



We know :
|a| = m
gcd(m,n) = 1

We want to show :
bn = a for some b in the group.

The Attempt at a Solution



Okay, I didn't really know where to start with this one, but I'll give it a try.

We know the gcd can be written as a linear combination, that is :

gcd(m,n) = 1 = ms + nt for some integers s and t.

Now :
1 = ms + nt
a1 = ams + nt
a = ams ant
a = easant
I'm not sure what you did to get the last line. a^{ms} = (a^m)^s = e^s = e, so the last line should be a = a^{nt}.
 
jbunniii said:
I'm not sure what you did to get the last line. a^{ms} = (a^m)^s = e^s = e, so the last line should be a = a^{nt}.

Ohhh whoops. I got confused by all the tags lol.

So really I have :

1 = ms + nt
a1 = ams + nt
a = ams ant
a = (am)sant
a = esant
a = ant

Hence a can be written as the nth power of some element in the group and we are done.
 
Zondrina said:
Ohhh whoops. I got confused by all the tags lol.

So really I have :

1 = ms + nt
a1 = ams + nt
a = ams ant
a = (am)sant
a = esant
a = ant

Hence a can be written as the nth power of some element in the group and we are done.


Right, specifically a = (a^t)^n.
 
jbunniii said:
Right, specifically a = (a^t)^n.

Yeah I wrote that down for my final solution after, I was just highlighting that your hint let me get there nice and quickly.

I wish I had analysis homework rather than this. I feel lost with algebra all the time.
 
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