Abstract Algebra question?

Question

How many elements are in :
U(9)?
U(25)?
U(p^2), for a prime?
U(p^k), for a prime and k a positive integer?

as much explanation as possible would be great!!

Answer 1

In general, the number of elements in U(n) is φ(n),
the number of natural numbers less than n and
relatively prime to n.
Here φ(n) is Euler's phi(totient) function.
Further φ(p²) = p(p-1) (*)
and φ(p^k) = p^(k-1)*(p-1).
Let's get an idea of why (*) works by showing that it's true.
Suppose we write the elements less than p² as follows
1 2...... p
p+1 p+2 2p
.,..................
(p-1)p + 1 (p-1)p + 2 ... p^2
Now count how many elements are multiples of p.
The only ones are the last elements of each row
and there are p of them.
So φ(p²) = p²-p = p(p-1).
The result for p^k is proved the same way.
Hope that makes sense!
So the number of elements in U(9) = 3*2 = 6
The number of elements in U(25) = 5*4 = 20.

Answer 2

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