/sci/ - Science & Math

I'm going through Lang's Basic Mathematics and got stuck on one of the problems regarding Isometry mappings.
The problem is in Chapter 6.4 - Multiplication Tables.
He proves that isometries for vertical and horizontal reflections are commutative but it is not proven for rotations and reflections.
He gives the solution HG^3 for row 2, column 5 in the picture, but from my understanding H o G doesn't equal HG^3.
The entire multiplication table idea seems fuzzy if you read the chapter.
Is this one of the errors people mention with regards to this book? I couldn't find anything about this chapter, only a youtube video where the author skips this section entirely.

I might not be able to respond immediately but I appreciate any input.