Groups

[Symmetry] => Actions

In [geometry]:

• [Reflection symmetry]
• [Rotational symmetry]
• [Translational symmetry]
• [Helical symmetry]
• [Scale symmetry] and [Fractal]
• [Glide reflection]

[Group]

In general, every kind of structure in mathematics will have its own kind of symmetry.

[Polynomial]

x^5 - x - 1 = (x - r0)(x - r1)(x - r2)(x - r3)(x - r4)

S5?

+ - x / √

[Noether's theorem]

[Conservation law] <=> Symmetry

[Momentum] <=> Translation in space

Energy <=> Translation in time

[Group action]

[Symbol]

Group table

[Isomorphism]

What are all the groups?

What are all the ways things can be symmetric?

Groups:

• Infinite Groups
• Finite Groups

Integers -break dowm-> Primes

Molecules -break down-> Atoms

Finite Groups -break down-> Simple Groups

• Find all the Simple Groups (18 infinite families + 26 leftovers (Sporadic Groups))
• Find all the ways to combine Simple Groups

Sporadic Groups:

• 20 Happy Family (Monster and its children)
  • Monster Group
  • Baby Monster Group
• 6 Pariahs

Monster Group:

• 1978, John McKay, Switch from Finite Group Theory into Galois Theory
• 1979, John Conway, Moonshine -> Relation between Monster and String Theory
• 1992, Richard Borcherds

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Reference: https://youtu.be/mH0oCDa74tE