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.