an introduction to group theory
DESCRIPTION
An Introduction to Group Theory. HWW Math Club Meeting (March 5, 2012) Presentation by Julian Salazar. Symmetry. How many rotational symmetries?. Symmetry. Symmetry. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/1.jpg)
AN INTRODUCTION TO GROUP THEORYHWW Math Club Meeting (March 5, 2012)
Presentation by Julian Salazar
![Page 2: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/2.jpg)
SYMMETRYHow many rotational symmetries?
![Page 3: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/3.jpg)
SYMMETRY
![Page 4: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/4.jpg)
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?
![Page 5: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/5.jpg)
SYMMETRY
Numbers measure size.
Groups measure symmetry.
![Page 6: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/6.jpg)
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: +, ×
![Page 7: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/7.jpg)
FORMAL DEFINITIONA 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 .)
![Page 8: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/8.jpg)
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?
![Page 9: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/9.jpg)
IDENTITIES ARE UNIQUETheorem: 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.
![Page 10: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/10.jpg)
INVERSES ARE UNIQUETheorem: 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 , .
![Page 11: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/11.jpg)
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 .
![Page 12: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/12.jpg)
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!
![Page 13: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/13.jpg)
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
![Page 14: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/14.jpg)
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
![Page 15: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/15.jpg)
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.
![Page 16: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/16.jpg)
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
![Page 17: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/17.jpg)
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
![Page 18: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/18.jpg)
LET’S “GROUP” OUR SYMMETRIES
What is What is ? What is What is ? . We’re not multiplying (not necessarily )
![Page 19: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/19.jpg)
CATEGORIZING GROUPSCyclic 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.
![Page 20: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/20.jpg)
ISOMORPHISMConsider 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.
![Page 21: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/21.jpg)
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!?
![Page 22: An Introduction to Group Theory](https://reader036.vdocuments.us/reader036/viewer/2022062301/56816245550346895dd2844a/html5/thumbnails/22.jpg)
USING GROUPSWith 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!