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Grouping is often important in TeX/LaTeX programming. I know of three ways to delimit a group:

  1. Via a \begingroup…\endgroup pair;
  2. Via a pair of matched braces {};
  3. Via a \bgroup…\egroup.

Now I am wondering: what is the difference between these three? Does their effect depend on whether they appear in math mode or text mode? In which way, if so?

Note: I am sure that if I look through the TeXBook, TeX By Topic, and some other resources I will find an answer. However, I am sure there are many others who are curious about this and do not have the time to engage in such a search -- just like me. It would surely be useful to have the info all in one place -- e.g. as an answer here.

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