Need help understanding Chapter 2 of Joseph J. Rotman's: An Introduction to Theory of Groups
Solution 1:
Is the former line just a typo, and if so, is it a known typo? Or am I missing something?
The use of $G$ should be $S$. Yes, it is known.
I always thought a subgroup was just a subset that satisfied the four group laws, am I mistaken?
There are multiple ways for a subset of a group to be a subgroup. Consider, for example,
Theorem (One-step Subgroup Test): Suppose $S$ is a nonempty subset of a group $G$. Then TFAE:
- $a,b\in S$ implies $ab^{-1}\in S$.
- $S$ is a subgroup of $G$.