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Group Theory

Basic idea:Numbers measure size, groups measure symmetry.” from Groups and Symmetry by Mark A. Armstrong

Why do we care about Symmetries?

One way to understand the intention of designer of an universe is to find the symmetries. The search for symmetry can be substantially different mental activity from the usual explicit calculations. For example, consider the following simple arithmetic: $$34126 \times 12378 - 12378 \times 34126 = ? $$ You can compute the first term using calculator, and store in the memory. The second term is calculated, and subtracted from the previous result stored in memory. The final numeric result tells us that one should use brain instead of fingers in scientific problems. Of course, the commutative property of multiplication is enough to write down the answer. The morale of this example is: find the greatest possible symmetry whenever possible

From Magnetic Monopoles in Grand Unified Theories by I. G. Koh

Physicists are mostly agreed that the ultimate laws of Nature enjoy a high degree of symmetry. By this I mean that the formulation of these laws, be it in mathematical terms or perhaps in other accurate descriptions, is unchanged when various transformations are performed. Presence of symmetry implies absence of complicated and irrelevant structure, and our conviction that this is fundamentally true reflects an ancient aesthetic prejudice - physicists are happy in the belief that Nature in its fundamental workings is essentially simple. Moreover, there are practical consequences of the simplicity entailed by symmetry: it is easier to understand the predictions of physical laws. For example, working out the details of very-many-body motion is beyond the reach of actual calculations, even with the help of computers. But taking into account the symmetries that are present allows one to understand at least some aspects of the motion, and to chart regularities within it.

The Unreasonable Effectiveness of Quantum Field Theory by R. Jackiw


A group $(G, \circ)$ is a set $G$, together with a binary operation $\circ$ defined on $G$, that satisfies the following axioms

  • Closure: For all $g_1, g_2 \in G$, $g_1 \circ g_2 \in G$
  • Identity: There exists an identity element $e \in G$ such that for all $g \in G$, $g \circ e = g = e \circ g$
  • Inverses: For each $g \in G$, there exists an inverse element $g^{-1} \in G$ such that $g \circ g^{-1}=e = g^{-1} \circ g $.
  • Associativity: For all $g_1, g_2, g_3 \in G$, $g_1 \circ (g_2 \circ g_3) = (g_1 \circ g_2) \circ g_3$.


We need a super-mathematics in which the operations are as unknown as the quantities they operate on, and a super-mathematician who does not know what he is doing when he performs these operations. Such a super-mathematics is the Theory of Groups.

Sir Arthur Stanley Eddington

Symmetry is the magic word that distinguishes theory from coincidence.

Seven Science Quests by Sudarshan

Almost anybody whose research requires sustained use of group theory (and it is hard to think of a physical or mathematical problem that is wholly devoid of symmetry) writes a book about it.


Matrix Groups and Matrix Algebras

To help your intuition further: Lie groups are “almost always” matrix groups as follows. There is a corollary to a difficult theorem known as Ado's theorem that every Lie algebra can be realized as a Lie algebra of square matrices. The same is not true of Lie groups: not every Lie group can be represented as a group of matrices but it is almost true (a consequence of the Peter-Weyl theorem is that every compact group can be realized as a group of square matrices). Certainly, since we can find a square matrix realization for every Lie algebra, we can build a matrix Lie group with that algebra as its Lie algebra through the matrix exponential function; then we find that matrix group's universal cover and this is where we sometimes fail to get a matrix group. This is not typical and the first Lie groups that were not also matrix groups (so called metaplectic groups) weren't found until 1937. These oddballs are all covering groups of noncompact groups.

By exponentiating the Lie algebra elements, which can always can be written as matrices, we get representations of the corresponding universal covering group that belongs to this Lie algebra.

Possible Points of Confusion?

Q: Why is the Jacobi identity important?

A: “[T]he real meaning of the Jacobi identity: it’s there so the adjoint representation of a Lie group, clearly a very basic and fundamental thing, induces a homomorphism in the corresponding Lie algebras that respects Lie brackets.”

Q: What does conjugation mean?



Important Groups

Group Invariants

Great Books

  • Visual Group Theory by Nathan Carter. From the preface: “If you are interested in learning about group theory in a relaxed, intuitive way, then this book is for you.
  • Naive Lie Theory by John Stillwell
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