Groups with the subtraction operation

Question

Why do integer mod integer sets with the operation of subtraction not form groups?

For example, integers mod 3 is {0,1,2}, which has an identity (0) and inverses (self inverses). And subtraction is an operation because any arguments into the operation outputs something still within integers mod 3. I suspect I am missing something as to why that is not a group.

Answer 1

Have you checked associativity?

For example, is $(2-1)-1=2-(1-1)$?

Answer 2

Subtraction is associative $\!\iff\! (x\!-\!y)\!-\!z = \overbrace{x\!-\!(y\!-\!z)}^{\textstyle x-y+z}\! \iff\! -z =z\iff\! \overbrace{x - y = x+ y}^{\!\!\!\!\!\text{subtraction}\,{\bf =}\, \text{addition}}$