Number Through a
Group Theoretic Lens
Hyman Bass
University of Michigan
Klein Project Session
Mathfest
Pittsburgh, August 7, 2010
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Euclidean Division,
or
Division with Remainder
Given a, b in R, a ≠ 0, there exist unique real q, r with
b
=
qa + r
q in Z
and 0 ≤ r ≤ |a|
-a
0
a
2a
. . .
(q-1)a
qa
b
(q+1)a
_.______.______._____.__ . . .____.______._____.___.______.____
|<- r ->|
q = qa(b) = [b/a]
and r = ra(b)
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Additive groups of (real) numbers
A set A of real numbers containing 0 and closed under + & -.
Examples:
R
Q
Z
-
Zm =
{qm | q in Z}
“Cyclic group”
e.g. {0}
-
Per(f) =
{ p in R | f(x+p) = f(x) for all x}
E.g. Per(sin)
=
Z2π
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Discrete Additive Groups
A is discrete if, for some ε> 0, |a| ≥ εfor all a ≠ 0 in A.
In this case, |a – b| ≥ εfor all a ≠ b in A.
Examples:
Cyclic groups are discrete
Q and R are not
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The Discrete/Dense Dichotomy
THEOREM An additive group of real numbers is either discrete
or dense in R
Suppose that A is not discrete. To show A is dense means that
given x in R and ε > 0, we can find a in A with |a – x| < ε. By
assumption there is a c ≠ 0 in A with |c| < ε. By Euclidean
Division, we can write x = qc + r with q in Z and 0 ≤ r < |c| < ε.
Then a = qc is in A, and |x – a| = r < ε.
COROLLARY If f(x) is continuous and non-constant, then
Per(f) is discrete.
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Inclusion and Divisibility
Equivalent conditions:
•b belongs to Za
•Zb ≤ Za
•b = qa for some integer q
Terminology and notation:
•a divides b
•b is a(n integer) multiple of a
•a | b
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Intersection and Sum
A and B are additive groups:
• A  B is an additive group, the largest one contained in
both A and B
•A + B = { a + b | a in A and b in B} is an additive
group, the smallest containing both A and B
•(The union is not an additive group unless one contains
the other)
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THEOREM Discrete additive groups are cyclic
In fact, if A is discrete, there is a unique a ≥ 0 such that
A = Za. If A = {0} take a = 0. Otherwise A contains
elements > 0, and then discreteness implies that there is
a least one, call it a. (Proof)
Then if b is in A, Euclidean Division gives b = qa + r
with 0 ≤ r < a. But r = b – qa belongs to A, so the
minimality of a > 0 in A implies that r = 0, so b = qa is
in Za.
Question: When is Za + Zb discrete?
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Commensurability
Equivalent conditions on non-zero real numbers a
and b:
•b belongs to Qa
•Qb = Qa
•a/b is rational
•a and b are both integer multiples of some d > 0
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Commensurability Theorem
Equivalent conditions on non-zero real numbers a and b:
1.Za + Zb is discrete
2.Za intersect Zb is not zero
3.a and b are commensurable
In this case, d = gcd(a, b) ≥ 0 and m = lcm(a b) ≥ 0 are
defined by
Zd =
Za + Zb
Zm =
Za intersect Zb
PROPOSITION ab = dm
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√2 is irrational, so . . .
•{a + b√2 | a, b in Z} is dense in R
•Diagonally cut the unit square into two triangles.
Starting at 0, lay copies of one end-to-end along the line
with a leg on the line, and lay copies of the other along
the line with the hypotenuse on the line. Consider the
sets of meeting points of each sequence of triangles.
Then they have no points in common, but come
arbitrarily close to each other.
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Properties of
and lcm
•If gcd(a, b) = 1 and a | bc, then a | c
•Primes and unique factorization theorem
•gcd(ca, cb)=
|c|gcd(a, b)
•gcd(a, b + qa)
=
gcd(a, b)
•Euclidean Algorithm to find gcd(a, b)
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Multiplicative groups
of (non-zero real) numbers
Examples:
Rx
Qx
aZ = {aq | q an integer}
{ ± 1}
(0, ∞)
(“Cyclic”)
What is a discrete multiplicative group?
M is discrete if there is an open interval around 1
whose intersection with M is {1}.
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New Phenomena
•Torsion: -1
•Direct products: Rx
•Homomorphisms:
=
{±1} x (0, ∞)
Sgn : Rx  {± 1}
| . |: Rx  (0, ∞)
Question: What are the discrete multiplicative
groups in Rx?
In (0, ∞)?
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THEOREM Discrete multiplicative
subgroups of (0, ∞) are cyclic
Proof: Continuous isomorphisms
exp: R

(0, ∞)
log: (0, ∞) 
R
M ≤ (0, ∞) discrete. A = log(M) is discrete in R,
hence A = Za.
Then
M = exp(A) = exp(Za) = mZ
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General discrete multiplicative groups
M a multiplicative group.
M>0 = M intersect (0, ∞)
1.M is discrete iff M>0 is discrete iff |M| is discrete
2.If -1 is in M then M>0 = |M| and M = {±1} x |M|
3.If -1 is not in M and M ≠ M>0, then M = aZ for some
a < 0.
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Discrete groups of Rotations
Let Rot denote the group (wrt composition) of
rotations of the plane. It consists of rotations ρ(a),
rotation about the origin by an angle of 2πa radians.
Thus we have a homomorphism
ρ: R  Rot
ρ(a+b) = ρ(a)oρ(b)
and ρ(Z) = I (= ρ(0)).
Let M be a discrete subgroup of Rot. Then it is easy
to see that A = ρ-1(M) is a discrete subgroup of R,
containing Z. Then A = Z•(1/n) for a unique n in N,
and so M is cyclic generated by ρ(1/n).
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Further Developments:
Additive and multiplicative groups of . . .
•C
(Connections with plane linear
algebra and geometry)
•Z/Zm
(Modular arithmetic, Chinese
Remainder Theorem, etc.)
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Thank you
[email protected]
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