What everyone asks about group theory
Group theory exists because order matters. Turn a square and then flip it, and you get a different result from flipping first. The subject is about what happens when $ab \neq ba$.
This guide is built from the questions thousands of people have actually asked about group theory, in the order a course meets them. Find the one that's bothering you, look at the real questions, then read the idea underneath them.
It assumes you've met the definition of a group. If you haven't, start here first. How I built it from Stack Exchange data is at the end.
A short notation corner
A lot of the traffic is people stuck on a symbol. The single biggest one, at 120,000 views, is someone asking what $\oplus$ means.
| $G \oplus H$, $G \times H$ | The same thing for finitely many groups: pairs $(g, h)$, multiplied one slot at a time. Abelian-group people write $\oplus$. |
| $(1\,2\,3)(1\,2)$ | Read right to left, like functions: first $(1\,2)$, then $(1\,2\,3)$. The answer is $(1\,3)$. Some books go left to right, which is the source of half the "my answer doesn't match" questions. |
| $D_4$ or $D_8$ | The symmetries of a square. Some authors count corners, some count elements. |
| $N \trianglelefteq G$ | $N$ is a normal subgroup of $G$. |
| $[G : H]$ | The index: how many cosets $H$ has in $G$. |
| $(\mathbb Z/n\mathbb Z)^\times$, $U(n)$ | The numbers mod $n$ that have a multiplicative inverse, under multiplication. |
| well defined | The answer doesn't depend on which name you used for the input. It matters whenever the inputs are cosets. |
Where these questions come from
I took the 2,000 most upvoted group theory questions on Math Stack Exchange and sorted them into chapters, in the order a course teaches them. I didn't expect the result. The biggest chapter by views, bigger than normal subgroups, homomorphisms or Sylow theory, is people trying to prove that some group is abelian: that $ab = ba$ after all.
The other line tells a story I'd already seen in linear algebra. Broadly, the further into the course, the more upvotes each view earns. The early chapters get the traffic from students with homework. The late ones get the appreciation from people who stayed. Same as there, "how do I" questions dominate the most viewed and "why" questions dominate the most upvoted, 32 to 7 and 6 to 14.