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=AB
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,
aG
• 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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