Context: In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in D4 above, once a reflection is performed, the square never gets back to the r2 configuration by just applying the rotation operations (and no further reflections), i.e. the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup H defines left and right cosets, which can be thought of as translations of H by arbitrary group elements g. In symbolic terms, the left and right cosets of H containing g are

Question: When is it preferable to consider two group elements to be the same?
Answer: irrelevant to the question whether a reflection has been performed

Question: What should not be considered when asking if a reflection has been performed?
Answer: rotation operations

Question: What number sets are used to show how subgroups can be seen as translations of the larger group?
Answer: Cosets

Question: When are two group elements considered different?
Answer: Unanswerable

Question: In D4 the square can return to the r2 configuration by applying only what?
Answer: Unanswerable

Question: What question are cosets irrelevant to?
Answer: Unanswerable

Question: What does group element g define?
Answer: Unanswerable

Question: When are rotation operations considered?
Answer: Unanswerable