So, I'm relearning Group Theory. And I got the axioms down, I think. So let's make a concrete example:
- The collection of numbers the positive integers less than 7: 1,2,3,4,5,6
- The • operation will be multiplication mod 7.
- Associativity holds.
- The Identity e is 1.
- Every element has an inverse:
- 1*? mod 7 = 1 --> 1
- 2*? mod 7 = 1 --> 4
- 3*? mod 7 = 1 --> 5
- 4*? mod 7 = 1 --> 2
- 5*? mod 7 = 1 --> 3
- 6*? mod 7 = 1 --> 6
But! What is the order of the group?! I thought the order would be 7. But there are 6 elements! So maybe I was wrong and 0 should be in the group.
But 0 does not have an inverse! There is no x such that 0*x mod 7 = 1.
So what am I misunderstanding here? Is it the definition of order? Is it some other trick about groups?