I've just started group theory and I don't know how to show this. Are you supposed to take examples of two elements from the group and an example modulo m (say 2 and 5, and modulo 3) and show the group axioms hold? That doesn't seem sufficient to me, it seems like I should be taking $a, b, c$ as elements of the group and proving it in general. I can see how to show that G has an identity element in that case but how do I show it has an inverse and that associativity holds?
Let $a, b, c \in G, G=(\mathbb{Z}/m\mathbb{Z})$
Identity:
$a * 1 = a$ so $1$ is the identity
Inverse:
I don't know how to show this…
Associativity:
I don't know how to show this either…
Best Answer
The problem is not well stated, as you cannot have $0$ in your group. It should ask that the integers in $\mathbb Z/m\mathbb Z$ that are coprime to $m$ be a group under multiplication. Associativity comes because it is true in $\mathbb Z$. To find the inverse you need Bézout's lemma