Exercise 3, Ch. 2, Sec. 3 - Algebra (Michael Artin): Let \(a,b\) be elements of a group \(G\), and let \(a' = bab^{-1}\). Prove that \(a = a'\) if and only if \(a\) and \(b\) commute.

Proof:

We have \(a = a' = bab^{-1}\). Multiplying on the right by \(b\) we get \(ab = ba(b^{-1}b) = ba\) and so \(a\) and \(b\) commute. This argument is completely reversible so we are done.

Exercise 2, Ch. 2, Sec. 3 - Algebra (Michael Artin): Prove that the products \(ab\) and \(ba\) are conjugate elements in a group \(G\).

Proof:

Define \(C_{x}:G\rightarrow G\) to be conjugation by the element \(x\). Then \[C_b(ab) = b(ab)b^{-1} = ba(bb^{-1}) = ba\] and \[C_a(ba) = a(ba)a^{-1} = ab(aa^{-1}) = ab.\] Thus \(ab\) and \(ba\) are conjugate elements in \(G\).

Exercise 1, Ch. 2, Sec. 3 - Algebra (Michael Artin): Prove that the additive group \({\mathbb{R}}^{+}\) is isomorphic to the multiplicative group \(P\) of positive reals.

Proof:

If you have ever tried to setup PostgreSQL on OpenBSD you may have run
into the following (very frustrating) error:

$ psql postgres
FATAL:  role "_postgresql" does not exist
psql: FATAL:  role "_postgresql" does not exist

I couldn't find any mention of this error in the pkg-readmes at
/usr/local/share/doc/pkg-readmes/postgresql-server-9.1.2p0 so I did
some serious searching through mailing lists and found the answer
which I'm reproducing here (from
http://www.mail-archive.com/misc@openbsd.org/msg68239.html) since

Just for practice with elisp I've decided to hit Project Euler and see how far I can get coding in elisp. Here's a solution to Problem 1 of Project Euler:

;;;
;;; Problem 1 of Project Euler: 
;;;

(defun modprod-three-or-five (x)
  "Returns the product of x mod 3 and x mod 5."
  (* (% x 3) (% x 5)))

(defun div-by-three-or-five (x)
  "Returns x if x is divisible by 3 or 5 otherwise returns 0"
  (if (eq (modprod-three-or-five x) 0) x 0))