Okay, you’re in abstract algebra. That’s an upper-level undergraduate math course, and that means you’re hot shit. All those mere mortals struggling through Calculus 3? Pathetic fools and, worse, engineers. It is shameful to think you were once like them, long ago, painstakingly computing the chain rule and alternately praying to and cursing the name of Saint Newton.
Except now you have to do all these impossible little algebra proofs that look easy and nobody will give you a straight answer about what you’re allowed to do, when, and what will make you look like the kid who still writes some letters backwards in 5th grade. Teachers will complain that you have to understand every little thing before using a rule.
Fuck that shit, there are proofs to be done and the Chinamen are coming to take our jorbs. I will teach your damn fool ass the rules of this game so you can score all the points and fuck all the math groupies.
Here are the useful manipulations you can do for any group operation.
Multiply both sides by something
a = b
ba = bb
Right-multiply both sides by something
a = b
ab^-1 = bb^-1
Cancellation on right and left
cad = cbd
a = b
Raise both sides of the equation to a power
ab = bc
(ab)^2 = (bc)^2
(ab)^2 = (bc)^3
abab = bcbcbc
Apply the socks-shoes property
(ab)^-1 = (bc)^-1
b^-1 a^-1 = c^-1 b^-1
Terms move freely through parentheses
(ab)c = de(fg)
abc = d(ef)g, etc.
(NB: a lot of the parentheses in our proofs are just there to make the substitutions more obvious. They are only strictly necessary when there are also exponents.)
If and only if the group is Abelian, you can flip terms around all willy-nilly.
ab = cde
ba = dec
If and only if the group is Abelian, you can distribute powers
(ab^2 c^-1 d^-2)^4 = a^4 b^8 c^-4 d^-8
“But how will I remember all this without memorizing?” Remembering is easy: just treat every operation as if it were multiplication, and drop the sign. Yeah, this means you don’t really know what each step means. You know when else that is true? Every time you ever did an algebra problem in your life.
This should make the algebra portion of your proofs quite manageable. Don’t forget you can throw in an ‘e’ (the identity element) whenever you feel like it too. Here are some problems to play around with:
1) a = a^2 b^-1. Solve for b.
2) (a (b^2))^-1 = (b^-1)^2. Solve for a^-1.
3) (a^-1 f g)^-1 c = (b d^-1 e^2)^2. Solve for a, assuming the operation is Abelian.
Let your normal algebra instincts guide you, but also remember to check the list after each step to make sure what you did was okay. Now the trick is just to figure out what you’re solving for and why! You will have to translate ideas like “closure” into algebra. Maybe that’s another post.