9.4ex I
Group 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!
 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.)
 Now, turn your informal proof of problem 1
into a PDI proof.
 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.
 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.
 Show that interpretation 4 really is a group. Explain
why these sorts of motions satisfy the three axioms of group theory.
 Explain in your own words why Interpretation
2 is not a group.
 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.
