What happens to the 0 element in a Finite Group? 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?

You're right, the group has order 6 because it has six elements.  You can make {0,1,2,3,4,5,6} a group with addition mod 7.  This would be a group of order 7.  
