I mean, I know intuitively what symmetry is.
But if we want to be exact, is it defined as a mapping from an object to itself preserving the structure? Or it is defined as a property that an object has, that is, the object remains unchanged under a set of operations?
Even the Wikipedia article about symmetry provides these two "definitions" one after another. But I cannot help but feel that they are different things. I would say that symmetry is property of invariance under a mapping, not the mapping itself.
For the context, I mainly work with symmetries in terms of groups. (I.e. rotational, reflectional symmetry etc.) Maybe this adds to my confusion, since there is a symmetric group $S_n$ with clear definition and operation, but also symmetry group (group of symmetries) which is a group of transformations of a given abstract object that preserve its structure.
In summary, how should be symmetry best defined/described?
Thank you for your insights on this.
May be relevant: A similar question on this forum, describing symmetry as a condition.