Noncommutative group with twelve elements.

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Homework Help Overview

The original poster seeks to identify a noncommutative group consisting of twelve elements. Initial attempts included a set of integers with subtraction as the operation, but this was challenged due to the set not containing twelve elements and failing to meet group criteria.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the validity of the original poster's approach, questioning the use of subtraction as a group operation and suggesting consideration of geometric groups related to symmetries.

Discussion Status

Some participants have provided feedback on the original poster's initial approach, pointing out its flaws and suggesting alternative directions. The discussion has evolved towards exploring geometric figures and their symmetries as potential candidates for a noncommutative group.

Contextual Notes

There is an emphasis on the need for a valid group structure, with specific attention to the properties of group operations and the requirement for twelve elements. The discussion includes considerations of geometric interpretations.

Reid
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Homework Statement



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


Homework Equations





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!
 
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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).
 
(1-2)-3=1-(2-3)? Subtraction isn't associative either. Subtraction is a pretty poor candidate for a group operation.
 
Thank you CompuChip! :)
 
And Dick :smile:

Did you solve it now?
 
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?
 
Reid said:
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?

That works fine.
 

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