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Clarifications about flag changes as a group action[edit source]

User:OfficialURL could you clarify what you meant by:

Generally, flag changes don't act as a group on the set of flags. They do however act as a group on the set of permutations of flags.

I don't understand what is meant here. Every group has at least one group action on every set, so the existence of group actions is not of any interests. I assume that this is saying it doesn't form a group action w.r.t. some implied operation? But I'm not sure what that operation is. It's also further confusing to me since I'm not sure what the intended group is on flag changes. The natural operation on flag changes, simple composition, forms a groupoid, rather than a group. You could can take this groupoid and identify all flag changes of the same rank to form a group, but these groups are kind of boring, have a really obvious action on flags, and the elements aren't really flag changes. To go beyond that I have said, I have no ideas. Thanks. Sycamore916 (talk) 21:22, 28 March 2023 (UTC)