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