Reflections - Understanding

Question

I have been asked

Does the set of all reflections in the plane form a group? Explain.

I thought that the answer was that it wouldn't as if I reflected the parabola $y=x^2$ through the y axis, I have a shape that is more of an "n" shape rather than a "u" shape.

However in my notes I have found that when listing the elements of a symmetry group of the same parabola, a reflection in the y axis exists.

Would someone be able to help clarify where I am going wrong? It could be with understanding the concept of a group, or what it means to reflect a function, or indeed both. Thanks for any assistance available

Answer 1

It depends on the group action. If the group action is composition (i.e. performing one reflection after another), then you have to answer four questions:

1. Is the identity operation (i.e. leaving the plane as-is) representable as one of the reflections in the set?
2. For any reflection $R$ you make, can you get back to the original function using another reflection (the inverse of $R$).
3. If you perform two arbitrary reflections in the set, $R_1$ and $R_2$, one after another, can the net result always be represented as a single reflection $R$ in the set?
4. Are compositions of reflections in the group associative?

If the answer to all questions is 'Yes', then the set of reflections you're looking at forms a group under composition.

Based on your initial description, assuming composition, the answer to your question is 'No', since leaving the plane as-is doesn't (typically) constitute a "reflection" and thus (1) is false.

As Dr. Kildetoft indicates, the answer depends on how exactly you define "reflection".