[james_s_tayler]

The integers form a group under multiplication though and a key property of a group is that every element has an inverse. So how does the definition hold if there is nothing that could be considered an inverse for the number 0? Curious about this... I never thought about it before.

[jacobolus]

The integers do not form a group under multiplication. As you noticed, the multiplicative inverse of any integer other than 1 or –1 is not an integer.

You might be thinking of the rational numbers (excluding zero).

[james_s_tayler]

Woops. You're right.

Looks like I need to play Group or Not Group.

https://youtu.be/qvx9TnK85bw

But then even for addition what's the inverse of 0? -0?

[ulrikrasmussen]

You are thinking of the multiplicative group of integers coprime to some integer n. That set by definition never includes 0.

[MrQuincle]

Integers form a monoid under multiplication.

PS: A monoid in which each element has an inverse is a group.