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10/11/2011
1
15251: Fall 2011
Algebraic Structures:
Groups Theory
Lecture 13 (October 11, 2011)
Il est peu de notions en mathematiques qui
soient plus primitives que celle de loi de
composition.
 Nicolas Bourbaki
Number Theory
Naturals
closed under +
a+b = b+a
a+0 = 0+a=a
Integers
closed under +
a+b = b+a
a+0 = 0+a=a
a+(a) = 0
Z
n
closed under +
n
a+
n
b = b+
n
a
a+
n
0 = 0+
n
a=a
a+
n
(a) = 0
(a+b)+c = a+(b+c)
(a+b)+c = a+(b+c)
(a+
n
b)+
n
c = a+
n
(b+
n
c)
A+0 =A
a+0 = a
a+(a) = 0
a+
n
0 = a
a+
n
(a) = 0
closed under *
closed under *
n
closed under *
(a+b)*c = a*c+b*c
ditto
ditto
A+(A) = 0
1/a may not exist
ditto
ditto
Number Theory
closed under +
A+B = B+A
closed under +
a+b = b+a
closed under +
n
a+
n
b = b+
n
a
(a+b)+c = a+(b+c)
(A+B)+C = A+(B+C)
Matrices
Integers
Z
n
(a+
n
b)+
n
c = a+
n
(b+
n
c)
A+0 = 0+A
a+0 = 0+a
a+(a) = 0
a+
n
0 = 0+
n
a
a+
n
(a) = 0
closed under *
closed under *
n
closed under *
(a+b)*c = a*c+b*c
ditto
ditto
A+(A) = 0
1/a exists if a
0
ditto
ditto
Number Theory
closed under +
A+B = B+A
closed under +
a+b = b+a
closed under +
n
a+
n
b = b+
n
a
(a+b)+c = a+(b+c) (a+
n
b)+
n
c = a+
n
(b+
n
c)
(A+B)+C = A+(B+C)
Invertible Matrices
Rationals
Z
n
(n prime)
Abstraction
:
Abstract away the inessential
features of a problem
=
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Today we are going to
study the abstract
properties of binary
operations
Rotating a Square in Space
Imagine we can
pick up the
square, rotate it
in any way we
want, and then
put it back on
the white frame
In how many different ways can we
put the square back on the frame?
R
90
R
180
R
270
R
0
F

F
—
F
F
We will now study these 8 motions,
called
symmetries of the square
Symmetries of the Square
Y
SQ
= { R
0
, R
90
, R
180
, R
270
, F

, F
—
, F
, F
}
Composition
Define the operation “
” to mean “first do
one symmetry, and then do the next”
For example,
R
90
R
180
Question: if a,b
Y
SQ
, does
a
b
Y
SQ
? Yes!
means “first rotate 90˚
clockwise and then 180˚”
= R
270
F

R
90
means “first flip horizontally
and then rotate 90˚”
= F
R
90
R
180
R
270
R
0
F

F
—
F
F
R
0
R
90
R
180
R
270
F

F
—
F
F
R
0
R
90
R
180
R
270
F

F
—
F
F
R
90
R
180
R
270
F

F
—
F
F
R
180
R
270
R
0
R
270
R
0
R
90
R
0
R
90
R
180
F
F
F

F
—
F
—
F

F
F
F
F
F
—
F

F
F
—
F
F
F

F
F
—
F
F

F

F
F
—
R
0
R
0
R
0
R
0
R
180
R
90
R
270
R
180
R
270
R
90
R
270
R
90
R
180
R
90
R
270
R
180
10/11/2011
3
How many symmetries for nsided body?
2n
R
0
, R
1
, R
2
, …, R
n1
F
0
, F
1
, F
2
, …, F
n1
R
i
R
j
= R
i+j
R
i
F
j
= F
ji
F
j
R
i
= F
j+i
F
i
F
j
= R
ji
Some Formalism
If S is a set, S
S is:
the set of all (ordered) pairs of elements of S
S
S
= { (a,b)  a
S and b
S }
Formally,
is a function from Y
SQ
Y
SQ
to Y
SQ
: Y
SQ
Y
SQ
→
Y
SQ
As shorthand, we write
(a,b) as “a
b”
“
” is called a
binary operation
on Y
SQ
Definition:
A binary operation on a set S is a
function
: S
S
→
S
Example:
The function f:
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This note was uploaded on 11/03/2011 for the course CS 251 taught by Professor Gupta during the Spring '11 term at Carnegie Mellon.
 Spring '11
 Gupta

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