Revisiting group theory

So, the 3d transformation operations (rotations / translations) on objects in 3d space are conveniently done using Lie algebra. I remember like 6-7 years back my high school teacher (and much more than that) suggested me to study Lie Algebra but there were no instance where I stumbled on it, in academics or in work. But very recently while studying 3D-Reconstruction, I encountered Lie group. So, I am planning to study it soon. Also, there this book Complex Analysis (Joseph Bak, Donald J. Newman) that was rusting back in my mind. And I am thinking to kindle it again.

I just realized that if I am not really comfortable with algebra stuffs like I am; more or less; in calculus stuffs.

Galois Fig: Galois (1811-1832, lived for only 20 years!) who is considered by many as the founder of group theory. He was the first to use the term “group” in a technical sense, though to him it meant a collection of permutations closed under multiplication. Galois was a mathematician and political activist, what a combination!

Group

Definition from Cayley (in 1884):

A set of symbols all of them different, and such that the product of any two of them (no matter in what order), or the product [operation] of any one of them into itself, belongs to the set, is said to be a group. These symbols are not in general convertible [commutative], but are associative.

Formal Definition: A group is a non-empty set G on which there is a binary operation \((a,b) \mapsto ab\) such that:

  • if a and b belong to G, then ab is also in G(closure),
  • $a(bc) = (ab)c$ for all a, b, c in G (associativity)

Last Updated: Tuesday, 4 Aug, 2020, 12:29 NPT
Author: Madhav Humagain (scimad)