Download ALGEBRAIC OBJECTS 1. Binary Operators Let A be a set. A

ALGEBRAIC OBJECTS
PAUL L BAILEY
L. MARIZZA A BAILEY
1. Binary Operators
Let A be a set. A binary operator on A is a function
∗ : A × A → A.
A binary operator is simply something that takes two elements of a set and gives
back a third element of the same set.
Example 1. Let R be the set of real numbers. Then + : R × R → R, given by
+(x, y) = x + y, is a binary operator. Also · : R × R → R, given by ·(x, y) = xy,
is a binary operator.
In general, in the sets N, Z, Q, R, and C, addition and multiplication are
binary operators. ¤
Example 2. Let X be a set and let P(X) be the power set of X. Then union
and intersection are binary operators on P(X); for example
∩ : P(X) × P(X) → P(X)
is defined by ∩(A, B) = A ∩ B, where A, B ⊂ X. ¤
Definition 1 (permutation). A permutation of a set X is a bijective function
σ : X → X. We call the set of all permutations of X, Sym(X).
Example 3. Let X be a set and let Sym(X) be the set of all permutations of
X. Then ◦ is a binary operator on Sym(X):
◦ : Sym(X) × Sym(X) → Sym(X)
is defined by ◦(φ, ψ) = φ ◦ ψ. ¤
Let A be a set and let ∗ : A × A → A be a binary operator. As in the
above examples, it is customary to write a ∗ b instead of ∗(a, b), where a, b ∈ A.
However, we keep in mind that ∗ is a function and that a ∗ b ∈ A.
Date: September 19, 2010.
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2. Closure
Let ∗ : A × A → A be a binary operator on a set A and let B ⊂ A. We
say that B is closed under the operation of ∗ if for every b1 , b2 ∈ B, we have
b1 ∗ b2 ∈ B.
Example 4. Let E be the set of even integers. Then E is closed under the
operations of addition and multiplication of integers. Indeed, the sum of even
integers is even, and the product of even integers is even.
Let O be the set of odd integers. Then O is closed under multiplication.
However, O is not closed under addition, because the sum of two odd integers is
even. ¤
√
Example 5. Let B = {a+b 2 ∈ R | a, b ∈ Q}. Then B is √
closed under addition
√
and multiplication of real numbers. For example, if a1 + b1 2 and a2 + b2 2 are
two element of B, then
√
√
√
(a1 + b1 2) + (a2 + b2 2) = (a1 + a2 ) + (b1 + b2 ) 2 ∈ B
and
√
√
√
(a1 + b1 2)(a2 + b2 2) = (a1 a2 + 2b1 b2 ) + (a1 b2 + a2 b1 ) 2 ∈ B.
Note that these results are in B because Q itself is closed under addition and
multiplication. Therefore a1 a2 + 2b1 b2 ∈ Q, and so forth. ¤
Exercise 1. In each case, we define a binary operation ∗ on R. Determine if
∗ is commutative and/or associative, find an identity if it exists, and find any
invertible elements.
(a) x ∗ y = xy + 1;
(b) x ∗ y = 21 xy;
(c) x ∗ y = |x|y .
Exercise 2. Consider the plane R2 . Define a binary operation ∗ on R2 by
x1 + x2 y1 + y2
(x1 , y1 ) ∗ (x2 , y2 ) = (
,
).
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Thus the “product” of two points under this operation is the point which is
midway between them. Determine if ∗ is commutative and/or associative, find
an identity if it exists, and find any invertible elements.
Exercise 3. Let I be the collection of all open intervals of real numbers. We
consider the empty set to be an open interval.
(a) Show that I is closed under the operation of ∩ on P(R).
(b) Show that I is not closed under the operation of ∪ on P(R).
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3. Groups
We will now study the objects called Groups. Given that so little is required
to have a group, they have quite an amazing structure. Theorems flow merely
from the definition of group and subgroup. We will study many examples of
groups and subgroups.
Definition 2. A group is a set G together with a binary operation · : G×G → G
satisfying:
(G1) g1 (g2 g3 ) = (g1 g2 )g3 for all g1 , g2 , g3 ∈ G (associativity);
(G2) there exists e ∈ G such that e · g = g · e = g for all g ∈ G (existence of
an identity);
(G3) for every g ∈ G there exists g −1 ∈ G such that gg −1 = g −1 g = e
(existence of inverses).
A group is a set that has an associative binary operation with an identity,
and in which every element has an inverse in the set.
Note that a set can be many different groups depending on the choice of binary
operation. Identify for which of the following binary operations, the sets below
are groups?
If the set is not a group under a binary operation, find the largest group in that
set under that operation.
(a) Z with multiplication? addition? concatenation?
(b) R with multiplication? addition?
(c) Q with multiplication? addition?
(d) F = {f : R → R} under function addition? function multiplication?
composition?
(e) SymX under composition? addition? multiplication?
(f ) The set of all strings in {0, 1} under concatenation? multiplication?
addition?
(g) Zn under addition? multiplication?
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There are two important theorems we need to get through. The first is that,
in a group,G, every element has a unique inverse. This allows the function
φ : G → G defined by φ(g) = g −1 to be well defined. The second is that their is
only one inverse. This allows the function which sends everything to the identity
to be well-defined.
Lemma 1 (Uniqueness of inverse). Let G be a group, and g ∈ G.
Let h ∈ G such that h ∗ g = e, and g −1 ∈ G such that g ∗ g −1 = e. Then
h ∗ g ∗ g −1 = h ∗ (g ∗ g −1 ) = h ∗ e = h.
Also h ∗ g ∗ g −1 = (h ∗ g) ∗ g −1 = e ∗ g −1 = g −1 .
Therefore, h = g −1 .
Lemma 2 (Uniqueness of identity). Let G be a group, with identity e. Suppose
there exists e0 ∈ G such that e0 ∗ g = g for all g ∈ G. Then e ∗ e0 = e by definition
of e0 identity,
and e ∗ e0 = e0 by definition of e identity.
Definition 3 (Abelian). A group is called abelian if its operation is commutative.
g∗h=h∗g
for all g, h ∈ G.
Example 6. The following are abelian groups:
•
•
•
•
•
•
•
(Z, +, 0), the integers under addition;
(Q, +, 0), the rational numbers under addition;
(R, +, 0), the real numbers under addition;
(C, +, 0), the complex numbers under addition;
(Q∗ , ·, 1), the nonzero rational numbers under multiplication;
(R∗ , ·, 1), the nonzero real numbers under multiplication;
(C∗ , ·, 1), the nonzero complex numbers under multiplication.
3.1. Standard Notation. Usually the operation of abelian groups is denoted
by +, and the identity is denoted as 0.
If the group is not abelian then the operation of the group is denoted as ·, and
the identity is denoted as 1 or e.
Example 7. Let X be a set and let P(X) = {A ⊂ X}. Define the symmetric
difference of A, B ∈ P(X) by
A4B = (A ∪ B) r (A ∩ B).
Then (P(X), 4, ∅) is a group under symmetric difference.
What is the identity?
What is the inverse of each element?
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Example 8. Let Mm×n (R) be the set of m × n matrices over the real numbers.
Then Mm×n (R) is a group under matrix addition.
What is the identity? What is the inverse of each element?
Example 9. Let GLn (R) be the set of invertible n × n matrices over the real
numbers. Then GLn (R) is a group under matrix multiplication. For those who
do not know GLn , a matrix is invertible if and only if it’s determinant is nonzero.
Finding the inverse of an arbitrary matrix can be time consuming, unless you
write a program that can do it for you. Let us examine GL2 R to identify the
identity and inverse of a matrix. We will then venture on to GL3 (R).
Example 10. Let X be a set and let Sym(X) = {σ : X → X | σ is bijective}.
Then (Sym(X), ◦, idX ) is a group under composition of functions. This is called
the permutation group of X. A permutation is defined to be a bijective function
from X to X.
Example 11. Let Cn = {g 0 , g 1 , g 2 , . . . , g n−1 } be a set of symbols, and define
a multiplication on Cn by g k g l = g k+l (mod n) . This is called a cyclic group of
order n . The identity is g 0 , and we write 1 = g 0 and g = g 1 .
• Cn is abelian;
• |Cn | = n;
Example 12. Let Sn be the set of permutations of the set {1, . . . , n}. The group
operation is composition of functions, and the identity is the identity function.
• Sn is nonabelian for n ≥ 3;
• |Sn | = n!.
Proof 1 ( Non-abelian). We know that f ◦ g does not always equal g ◦ f . In
order to prove that Sn is not abelian for n ≥ 3, we are going to have to find two
permutation of 3 things which do not commute.
Let (123) be the permutation which sends 1 → 2, 2 → 3 and 3 → 1.
and (12) be the permutation which sends 1 → 2 and 2 → 1.
Then (123)(12) = (13) because 1 → 2 → 3 and 2 → 1 → 2 and 3 → 3 → 1.
However (12)(123) = (23).
Since every permutation group of n things where n ≥ 3 must also permute 3
things, then these elements are contained in all larger permutation groups.
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Example 13. Let An be the set of even permutations of the set {1, . . . , n}; that
is, An is the set of permutations whose decomposition into transpositions always
yields an even number of transposition.
• An is nonabelian for n ≥ 4;
• |An | = n!/2.
• An ≤ Sn ;
Example 14. Let Dn be the set of symmetries of a regular n-gon (n ≥ 3); that
is, Dn is the set of rigid motions of a regular polygon with n-vertices which send
vertices to vertices. Then Dn is generated by a rotation and a reflection. For
example, consider an equilateral triangle. We may rotate it by 120 degrees, 240
degrees, or 360 degrees; the latter represents the identity map. We may also
reflect it through a line which intersects a vertex and bisects the opposite side.
Department of Mathematics, Basis Scottsdale
E-mail address: [email protected]