The universe is an enormous direct product of representations of symmetry groups.
—Steven Weinberg
群是抽象代數裡面很重要的概念，它是描述對稱性的數學語言。
Late 1700s- Joseph-Louis Lagrange (1736-1813) 利用置換的概念，理解了三次和四次方程為什麼有解。（Paolo Ruffini利用同樣的思想證明了一般的五次或以上方程沒有根式解）
Early 1800s-Évariste Galois (killed in a duel in 1832 at age 20), and Niels
Abel (died in 1829 at age 26 of TB) 闡述了代數方程的可解性和置換群的聯繫，真正利用群的概念解決了這個難題。
第一個正式的群的概念由Cayley於1854提出。Cayley, Hamilton, and Sylvester 引進了矩陣的表達。1890年Fedorov發展了各種對稱性操作的數學方法，證明了晶體有且僅有230種空間群，次年Schönflies也獨立的證明了這個結果。1920年群論開始應用於物理、化學等領域。裡面還有很多著名數學家和物理學家的貢獻，不一一而舉。
群論在物理和化學中究竟有啥用？
It is often hard or even impossible toobtain a solution to the Schrodinger equation-however, a large part ofqualitative results can be obtained by group theory. Almost all the rules of spectroscopy follow from the symmetry of a problem. -Eugene Wigner (1963 Nobel Prize in Physics)
1. 群論在量子力學中的應用
舉兩個很優雅例子：
a. 能級簡併度的確定
Each eigenvalue of a Hamiltonian there corresponds a unique irreducible representation of the group of that Hamiltonian. The degeneracy of an eigenvalue is the dimensionality of this irreducible representation. Thus, the dimensionalities of the irreducible representations of a group are the possible degeneracies of Hamiltonians with that symmetry group.
b. 布洛赫定理
固體物理中最重要的結論之一，完全可以由群論得出，其證明亦簡單。
定義一個平移操作
可知Hamiltonian是一個N階的阿貝爾循環群。
Character table如下：
便可證明
更多信息參考：
Tinkham, Group Theory and Quantum Mechanics(1964)
2. 選擇定則（Selection Rules）
3. Molecular Vibrations, Infrared, and Raman Activity.
4. 能帶理論，k dot p model
5.
未完待續，長期更新。。。
Ref:
"Group Theory: Applications to the Physics of Condensed Matter" by M. S. Dresselhaus
「Chemical Applications of Group Theory」 by F. Albert Cotton
「Molecular Symmetry and Group Theory,」 by Carter
"Group Theory and Quantum Mechanics" by Tinkham
化學相關書籍（Hughbanks推薦）：
Author(s),Title
Ballhausen, Introduction to Ligand Field Theory
Ballhausen, Q. M. and Chemical Bonding in Inorganic Complexes
Bishop, Group Theory and Chemistry
Burdett, Molecular Shapes
Albright, Burdett, Whangbo, Orbital Interactions in Chemistry, 2nd Edition
Burns & Glazer, Space Groups for Solid State Scientists
Butler, Point Group Symmetry Applications
Figgis, Introduction to Ligand Fields
Flurry, Symmetry Groups
Franzen, Physical Chemistry of Solids, Basic Principles …
Hanna, Quantum Mechanics in Chemistry
Hoffmann, Solids and Surfaces, A Chemist’s View…
Heine, Group Theory in Quantum Mechanics
Kettle, Symmetry and Structure
Lax, Symmetry Principles in Solid State and Molecular Physics
McQuarrie & Simon, Physical Chemistry
Molloy, Group Theory for Chemists
Murrell, Kettle & Tedder, The Chemical Bond
Pearson, Symmetry Rules for Chemical Reactions
Walton, Beginning Group Theory for Chemistry