# Noncommutative group with twelve elements.

1. Oct 22, 2008

### Reid

1. The problem statement, all variables and given/known data

I need to find a group with twelve elements that is noncommutative.

2. Relevant equations

3. The attempt at a solution

I was considering a group G of the integers (-5,-4,-3,-2-,-1,0,1,2,3,4,5) with respect to subtraction. Subtraction is associative/noncommutative, G has the neutral element, and every element has an inverse! Excellent!

But (!) it composes only eleven elements! I need twelve!

2. Oct 22, 2008

### CompuChip

Also, it is not a group.
3 - (-3) is not an element of your set.
0 is not a left identity element either: 0 - 3 is not 3.

Try thinking about geometric groups (symmetries of 2 and 3-dimensional objects).

3. Oct 22, 2008

### Dick

(1-2)-3=1-(2-3)??? Subtraction isn't associative either. Subtraction is a pretty poor candidate for a group operation.

4. Oct 22, 2008

### Reid

Thank you CompuChip! :)

5. Oct 22, 2008

### CompuChip

And Dick

Did you solve it now?

6. Oct 22, 2008

### Reid

At least I think I did. :) I regarded the geometrical figure with six edges. And looked at the mirroring and rotations of that structure. Is that better?

7. Oct 22, 2008

### Dick

That works fine.