I'm having a hard time grasping groups in Group Theory. Is it okay to visualise them as being sets with the group axioms and a binary operation, intuitively as a Venn diagram? Also, $(G,*)$ and $G$ without $*$ is really confusing me. I can't seem to move on in my study of Group Theory because of this. Can anyone also recommend me any online sources for a clear understanding of groups, please? What are some applications of groups? How can I apply them and see them to better understand their purpose. I feel really anxious whenever I open my textbook on Group Theory and it's because of these answers that I'm missing. I've looked through countless books and online notes. Everything is seems too complex and doesn't sit in my mind. I would love to study this subject more effectively. I had this problem while studying Set Theory and I need another approach for Group Theory. Thank you!
[Math] How to visualise groups in Group Theory
abelian-groupsbinary operationsgroup-theoryreference-request
Best Answer
Groups are symmetries. The best way to think about a group is that you have an object that has some kind of symmetry, and the group represents mathematically that symmetry. For example, $D_8$ is the symmetry of a square, with each element representing a rotation of flip of the square; $C_5$ is the symmetry of five objects rotating in a circle, with each element representing a different rotation; and $S_7$ is the symmetry of $7$ points that can be permuted in any fashion, with each element corresponding to a different reording.
In my opinion, this is far and away the best way to think about groups, even abstractly. You have some unspecified object and each element of the group represents a way that an object or collection of objects can be spun, flipped, reflected, etc. without changing.
See here for a cute interactive illustration of $D_8$ being the symmetry group of a square.