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

[STEMucator]

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 = a^{ms + nt}
a = a^ms a^nt
a = ea^sa^nt
a = a^sa^nt

Here's where I get stuck. Any pointers?

 

[jbunniii]

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 = a^{ms + nt}
a = a^ms a^nt
a = ea^sa^nt

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 $a = a^{nt}$.

 

[STEMucator]

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 <sup>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.</sup>

 

[jbunniii]

Ohhh whoops. I got confused by all the <sup>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.</sup>

^{Right, specifically $a = (a^t)^n$.}

 

[STEMucator]

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.