#### 9.4ex IGroup Theory Proofs

Derivations are a little too difficult to do on the computer at this stage. So, try to write out proof to the following the old fashioned way: on paper!

Some of these require that you follow the dictates of PLIF and PDI. Others you may want to do less formally. Use the tutorials examples to give you an idea of how to proceed. Then try to have fun!

1. Theorem 1 says that if e and f are identity elements, then e=f. But before we pick 'e' as our name of the one and only identity element, we need to prove that there is exactly one identity element. The proof is easy from theorem 1 and one of the axioms of group theory. But you should state it informally. (Feel free to take theorem 1 as a premise of your proof.)
2. Now, turn your informal proof of problem 1 into a PDI proof.
3. Tutorial four ends with the theorem that there can be only one inverse for an element of a group. That's the informal statement. Give the statement of this theorem more formally in our language PLIF.
4. It's a bit long, but try to give the derivation which proves your statement from 1 using the rules of PDI in using the language PLIF.
5. Show that interpretation 4 really is a group. Explain why these sorts of motions satisfy the three axioms of group theory.
6. Explain in your own words why Interpretation 2 is not a group.
7. All the groups we studied in the tutorial satisfy this additional axiom:

(^x)(^y)I{x+y}{y+x}

All such groups are called "abelian". Now consider a new interpretation: Interpretation 5. This interpretation is just like the motions of Interpretation 4 except that it includes projections onto a plane. A projection onto a plane just takes a two dimensional object in 3 space and transforms each point of the object not on the plane to the point on the plane closest to the original. Show that the collection of motions generated by translations, rotations and projections is not abelian.