And finally, if a symmetry preserves something, then doing it in reverse should also be a symmetry (so it has an inverse). The identity map always preserves everything. If one symmetry preserves something, then composing two symmetries also preserves it (so symmetries should be "closed under composition"). Well, "acting" on something means it's a map of some sort, so composition of symmetries should be associative since composition of maps is associative. A "symmetry" is something that acts on something in such a way that it "preserves" some property. On some level, the definition of a group is really just an abstraction of the intuitive idea of what a "symmetry" is. I'm not familiar with Herstein, but some books certainly push the idea of groups as "symmetries" more than others. If you're successfully doing the exercises, then you probably understand more than you think.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |