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
HWW Math Club Meeting (March 5, 2012)
Presentation by Julian Salazar
How many rotational symmetries?
Numbers measure size.
Groups measure symmetry.
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):
The integers under addition are a group.
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.
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 is a binary operator.
Take functions and .
We compose and to get
, so we evaluate from right to left.
means first, then .
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.
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.
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!?
With groups you can: