Hints for Homework #1
When you take a GCD or LCM of integers that are already
factored into primes, you can look at the exponents of the
prime factors to get the answer, rather than going through
the whole Euclidean algorithm.
The final exercise of the homework asks you to compute
FRFR as a permutation. By this I mean to use the presentation
of R and F in the tabular permutation notation I introduced
in Lecture 2, not the geometric representation as symmetries
of the square. Simply follow the permutations in order
from left to right.
Start by computing FR, then apply F to the result to get FRF,
then finally apply R to the result to get FRFR.
(Please note that if you read further in Chapter 5, you'll
find that Gallian treats composition of permutations
differently. I'll cover this in lecture, but it is really
only a notational difference, so don't worry about this
difference now.)
Chapter 2:
For the "shoes and socks" problem #16, try using a dihedral
group to construct the counterexample called for. You
simply need two elements a, b in some group for which
the stated relationship does not hold.
The inverse of (ab)^2 = abab is (b^-1 a^-1)^2, which
is not necessarily equal to b^-2 a^-2.
For #18, use the defining property of the inverse to prove
this relationship. Note that (a^-1)^-1 is not the
same as (a^-2), which equals (a^-1)^2.
Take the defining property of the inverse
(for every g in G, there is an inverse g^-1 for which
g g^-1 = g^-1 g = e), substitute g = a to get one
equation, then substitute g = a^-1 to get a second equation.
Then cite a uniqueness theorem (by name and/or number)
to show that two inverses must be identical.