I'm told that symmetries in hyperbolic space is - I'm not sure what that is!
I'm assuming o is a flip and r is a rotation. Meaning:
r( 123 ) → 231
o( 123) → 321
The great thing about it is that this is the set of symmetries of a triangle. r represents a rotation, o represents a flip over an altitude of the triangle.
o and r are the generators of the group:
So to find these inverses, you basically just do what you did, backwards, in the opposite order.
I'm using ' as the inverse symbol.
Start with the following:
o² = e
r³ = e
o' = o
r' = r²
r² ' = r
e is the identity
This is because o has order 2 and r has order 3. See:
Using those rules, we get:
(or)' = r'o' = r'o = r²o
o'r' = or' = or²
r'o' = r²o
What's nice about these is that if you rotate, then flip, you could have flipped, then rotated in the opposite direction. Does that make sense? So:
(or)² = oror = oor'r = o²e = o² = e
You can test that out if you don't believe it:
123 → 321 → 213 → 312 → 123
o²r² = er² = r²