An introduction to group theory
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An Introduction to Group Theory. HWW Math Club Meeting (March 5, 2012) Presentation by Julian Salazar. Symmetry. How many rotational symmetries?. Symmetry. Symmetry.

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An Introduction to Group Theory

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An introduction to group theory

An Introduction to Group Theory

HWW Math Club Meeting (March 5, 2012)

Presentation by Julian Salazar


Symmetry

Symmetry

How many rotational symmetries?


Symmetry1

Symmetry


Symmetry2

Symmetry

  • So the 6-pointed snowflake, 12-sided pyramid, and the regular tetrahedron all have the same number of rotational symmetries…

  • But clearly the “nature” of their symmetries are different!

  • How do we describe this?


Symmetry3

Symmetry

Numbers measure size.

Groups measure symmetry.


Terminology

Terminology

  • Set: A collection of items. Ex:

  • Element: An item in a set. Ex:

  • Binary operator: An operation that takes two things and produces one result. Ex: +, ×


Formal definition

Formal Definition

A group is a set with a binary operation . We can notate as or simply . We can omit the symbol.

It must satisfy these properties (axioms):

  • Closed: If , is in

  • Associative:

  • Identity: There exists such that for every in . ()

  • Inverse: For every in here exists such that .)


Example

Example: (,+)

The integers under addition are a group.

  • Closed: Adding two integers gives an integer

  • Associative: It doesn’t matter how you group summands; (3+1)+4 = 3+(1+4)

  • Identity: Adding an integer to 0 gives the integer

  • Inverse: Adding an integer with its negative gives 0

  • Is , ×) a group?

  • Is , ×) a group?


Identities are unique

Identities are UNIQUE

Theorem: For any group, there’s only one identity .

Proof: Suppose are both identities of .

Then .

Ex: For (,+), only 0 can be added to an integer to leave it unchanged.


Inverses are unique

Inverses are UNIQUE

Theorem: For every there is a unique inverse .

Proof: Suppose are both inverses of .

Then and .

Ex: In (,+), each integer has a unique inverse, its negative. If , .


Composition a note on notation

Composition (A note on notation)

Composition is a binary operator.

Take functions and .

We compose and to get

, so we evaluate from right to left.

means first, then .


Let s group our symmetries

Let’s “Group” our Symmetries

  • Let be doing nothing (identity)

  • Let the binary operation be composition

  • Let be rotation 1/12th clockwise

  • So means…

  • rotating th of the way clockwise!


Let s group our symmetries1

Let’s “Group” our Symmetries

  • Closed: No matter how many times you rotate with , you’ll end up at another rotation

  • Associative: Doesn’t matter how you group the rotations

  • Identity: You can do nothing

  • Inverse: If you rotate by , you must rotate to return to the original state


Let s group our symmetries2

Let’s “Group” our Symmetries

  • What is rotating counter-clockwise then?

  • It’s (the inverse of ).

  • But wait! .

  • So . In plain words, rotating 1/12th twelve times gets you back to where you started =O


Let s group our symmetries3

Let’s “Group” our Symmetries

  • But inverses aren’t unique! , right?

  • Yeah, but the net effect of is the same as that of .

  • So in our set, we only have .

  • We only count “unique” elements for our group.


Let s group our symmetries4

Let’s “Group” our Symmetries

  • Let be doing nothing (identity)

  • Let the binary operation be composition

  • Let be rotating the sign 1/8th clockwise

  • Let be flipping the sign over


Let s group our symmetries5

Let’s “Group” our Symmetries

  • Closed: No matter which rotations you do, you’ll end up at one of the 16 rotations

  • Associative: Doesn’t matter how you group the rotations

  • Identity: You can do nothing

  • Inverse: You can always keep rotating to get back to where you started


Let s group our symmetries6

Let’s “Group” our Symmetries

  • What is

  • What is ?

  • What is

  • What is ?

  • .

  • We’re not multiplying (not necessarily )


Categorizing groups

Categorizing Groups

Cyclic groups: Symmetries of an -sided pyramid.

Note: (,+) is an infinite cyclic group.

Dihedral groups: Symmetries of a -sided plate.

Other:

Permutation groups: The permutations of elements form a group.

Lie groups, quaternions, Klein group, Lorentz group, Conway groups, etc.


Isomorphism

Isomorphism

Consider a triangular plate under rotation.

Consider the ways you can permute 3 elements.

Consider the symmetries of an ammonia molecule.

They are ALL the same group, namely

Thus the three are isomorphic.

They fundamentally have the same symmetry.


Groups are not arbitrary

Groups are NOT arbitrary!

Order: # of elements in a group

There are only 2 groups with order 4:

,

There are only 2 groups with order 6:

B 1 group with order 5!?


Using groups

Using Groups

With groups you can:

  • >>Measure symmetry<<

  • Express the Standard Model of Physics

  • Analyze molecular orbitals

  • Implement public-key cryptography

  • Formalize musical set theory

  • Do advanced image processing

  • Prove number theory results (like Fermat’s Little Theorem)

  • Study manifolds and differential equations

  • And more!


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