[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?