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.

See: http://en.wikipedia.org/wiki/Dihedral_gr…

o and r are the generators of the group:

http://en.wikipedia.org/wiki/Generating_…

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:

http://en.wikipedia.org/wiki/Order_%28gr…

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

And finally:

o²r² = er² = r²

herpes testing