Transcript Groups

Groups
Physics and Mathematics

Classical theoretical physicists were often the
preeminent mathematicians of their time.
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Descartes – optics; analytic geometry
Newton – dynamics, gravity, optics; calculus, algebra
Bernoulli – fluids, elasticity; probability and statistics
Euler – fluids, rotation, astronomy; calculus, geometry,
number theory
• Lagrange – mechanics, astronomy; calculus, algebra,
number theory
• Laplace – astronomy; probability, differential equations
• Hamilton – optics, dynamics; algebra, complex numbers
Sets

Set notation
• Set X = {x: P(x)}

Union and intersection
• X  Y = {x: x  X or x  Y}
• X  Y = {x: x  X and x  Y}

A
C
Subset
B
• Y  X ,if  y  Y, then y  X

Cartesian product
• X  Y = {(x, y): x  X, y  Y}
C=AB
Map

A map is an association from
one set to another.
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
Sets X = {x}, Y = {y}
Map f: X  Y
X is the range
Y is the domain
X
f
Maps are also called
functions.
Y
• f: X  Y or x  f(x)
x  X, f(x)  Y
Image

Functions define subsets
called image sets.
• f(X) = {f(x); x  X}

X
Injective or one-to-one:
• Any two distinct elements of
X have distinct images in Y.
•  x1, x2  X, where x1 ≠ x2,
then f(x1) ≠ f(x2).

Surjective or onto:
• The image of X under f is the whole of Y.
•  y  Y,  x  X, such that f(x) = y.
f
f(X)
Y
Binary Operation

A binary operation on a set A
is a map from A  A  A.

• f(a,b) = a ◦ b = c; a, b, c  A

• Addition is both
• Subtraction is neither
Associative operation:
• a ◦ (b ◦ c) = (a ◦ b) ◦ c

Commutative operation:
• a◦b=b◦a
Binary operations
on the real
S1
numbers R may be
associative and
commutative.

Matrix multiplication is
associative, but not
commutative.
Group Properties

Groups are sets with a
binary operation.
• Call it multiplication
• Leave out the operator sign

Group definitions: a, b, c  G
• Closure: ab  G
• Associative: a(bc) = (ab)c
• Identity:  1  G, 1a = a1 = a,
aG
• Inverse:  a-1  G, a-1a = aa-1
= 1,  a  G
Problem
 Are these subsets of Z, the
set of integers, groups under
addition?
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Z+: {n: n  Z, n > 0}
even numbers: {2n: n  Z}
odd numbers: {2n+1: n  Z}
{±n2: n  Z}
{0}  {±2n : n  Z+}
Discrete Group

a
a b c d
a b c d
b
c
d
b
c
d
c
d
a
d
a
b
a
b
c

A table can describe
a group
S1
with a finite number of
elements.

Repeated powers of b
generate all other elements.
• A cyclic group
• b is a generator
– b2 = c
– b3 = d
– b4 = a
Isomorphism

1
1 i 1  i
1
i 1  i
i
i 1  i 1
1 1  i 1
i
i i 1
i 1
The complex units are isomorphic
to the cyclic 4-group.

A group may have
other
S1
ways of realizing the
elements and operation.

If the realization is one-toone and preserves the
operation it is isomorphic.

A homomorphism preserves
the operation, but is not oneto-one.
Matrix Representation

Groups are often
represented by matrices.
• Unitary matrices with
determinant 1

The elements of any finite
group can be represented by
unitary matrices.
• Also true for continuous Lie
groups
1 0

A  
0
1


 0  1

B  
1
0


 1 0 

C  
0

1


 0 1

D  

1
0


These matrices are also
isomorphic to the cyclic 4-group.
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