Prove that that $U(n)$, which is the set of all numbers relatively prime to $n$ that are greater than or equal to one or less than or equal to $n-1$ is an Abelian group.
My thought process: for $a, b \in U(n)$
Associativity: $(a + b) + c = a + (b + c)$
Identity: $1$ is in the set so $a\cdot 1 = a = 1\cdot a$
Inverse: I'm stuck on how to determine the inverse of the set if it exist.
Abelian criteria : $a\cdot b = b\cdot a$
Thanks
Best Answer
It’s true that you know that multiplication in $\Bbb Z$ is associative and commutative, but you still have to prove that multiplication in $U(n)$ is associative and commutative, i.e., that multiplication modulo $n$ is associative and commutative. To show that every element of $U(n)$ has a multiplicative inverse in $U(n)$, use Bézout’s lemma: if $a$ and $n$ are relatively prime, there are integers $u$ and $v$ such that $au+vn=1$.