Start here: group theory

These guides are a companion to learning group theory, not a whole course. This page covers what you should know first, what to read alongside, and which guide to open when.

Are you ready?

Group theory needs less background than people expect. There's no calculus, and linear algebra only turns up in a few examples. What it does need is comfort with functions and with short proofs. Try each of these before opening the answer.

clock arithmetic

What is $17 \bmod 5$? What is $(4 \times 3) \bmod 5$?

answer

Both are $2$. "Mod 5" means the remainder after dividing by 5, like a clock with 5 hours. The integers mod $n$ are the first groups you'll meet.

functions

On the whole numbers, is $f(n) = 2n$ one-to-one? Is it onto?

answer

One-to-one, yes: different inputs give different outputs. Onto, no: odd numbers are never hit. Permutations, the stars of the subject, are the functions that are both.

order matters

With $f(x) = x + 1$ and $g(x) = 2x$, is $f(g(x))$ the same as $g(f(x))$?

answer

No: $f(g(x)) = 2x + 1$ but $g(f(x)) = 2x + 2$. That's group theory in miniature. Most of the subject is about what happens when the order you do things in matters.

proof, for the problems page

Prove that if $n^2$ is even, then $n$ is even.

answer

Prove the contrapositive: if $n$ is odd, $n = 2k + 1$, and $n^2 = 4k^2 + 4k + 1$ is odd.

If you couldn't write something like that, read the first few chapters of Richard Hammack's free Book of Proof before the problems page, especially the ones on contrapositive and contradiction.

Which guide to open

Brand new
Start with the field guide. Read the first three families and click the plates until "every row contains every colour once" feels obvious. Then work through the problems alongside a book, one part at a time.
Took a course, still shaky
Go to what everyone asks and jump to your chapter. For most people it's normal subgroups and quotients.
Want it to stick
Do the problems with the scratchpad open, and build every example you're unsure about in it before trying to prove anything.

What to read alongside

Nathan Carter, Visual Group Theory. Pictures first, like the field guide. His free Group Explorer draws Cayley tables and diagrams for you.
Charles Pinter, A Book of Abstract Algebra. Cheap, gentle and conversational. The best choice if you've never read a proof-based book.
Thomas Judson, Abstract Algebra: Theory and Applications. Free and complete. The chapter names below are from this book.
Joseph Gallian, Contemporary Abstract Algebra. The most common course textbook, with lots of exercises and examples.

How the pieces line up

Each part of the problems page, with where to read about it and where it turns up in the other two guides.

problems	Judson	questions guide	field guide
Moves	Groups	what a group is	$S_3$ and $D_4$
Order	Cyclic Groups	orders and generators	the cyclic family
Cosets	Cosets and Lagrange's Theorem	subgroups and Lagrange	$A_4$
Permutations	Permutation Groups	permutations	$S_3$, $S_4$
Maps and quotients	Isomorphisms; Normal Subgroups and Factor Groups; Homomorphisms	homomorphisms, quotients	$D_4$ against $Q_8$
Commuting	not covered	proving it commutes	the commuting view of any plate
Actions	Group Actions	actions and counting, conjugacy	$S_4$, the cube
The finale	Normal Subgroups and Factor Groups (simplicity of $A_n$)	Sylow and small groups	$A_5$

Permutations come fourth here but earlier in some books. It doesn't matter much: the Permutations part only needs the Moves part.