Abstract Algebra

To show that binary structures are isomorphic:

1. Define the function ∅ that gives the isomorphism of S with S'. (Define what ∅(s) is to be for every s in S.) 2. Show that ∅ is a one-to-one function. (Suppose that ∅(x) = ∅(y) in S' and deduce that x=y in S.) 3. Show that ∅ is onto S'. (Suppose that s' ∈ S' is given and show that there does exist s ∈ S such that ∅(s) = s'.) 4. Show that ∅ (x * y) = ∅(x) *' ∅(y) for all x, y ∈ S. (Computation: compute both sides of the equation and check to see if they are equal.)

To define a binary operation on a set S:

1. Exactly one element is assigned to each possible ordered pair of elements in S. 2. for each ordered psi of elements in S, the element assigned to it is again in S.

Binary Algebraic Structure

A binary algebraic structure <S,*> is a set S together with a binary operation * on S.

Binary Operation

A binary operation * on a set S is a function mapping S x S into S. For each (a,b) ∈ S x S, we will denote the element * ((a,b)) of S by a * b. Ex. Addition and Multiplication on binary operations on R Z C R+ Z+ Let M(R) be the set of all matrices with real entries. Matrix addition (+) is not a binary operation b/c if A and B have different sizes, they can not be added.

Associative

A binary operation * on a set S is associative if (and only if) (a * b) * c = a * (b * c) for all a, b, c ∈ S.

Commutative

A binary operation * on a set S is commutative if (and only if) a * b = b * a for all a, b ∈ S.

Group

A group <G, *> is a set G, closed under a binary operation *, such that the following axioms are satisfied: A1. Closure G1. (A2) Associativity ∀ a,b,c ∈ G, we have (a*b)*c = a*(b*c), associativity of * G2. (A3) There is an element e in G such that ∀ x∈G, e*x=x*e=x, identity element e for x. G3. (A4) Corresponding to each a ∈ G, there is an element a' in G such that a*a'=a'*a=e, inverse a' of a.

Element

A set S is made up of elements, and if a is one of these elements, we shall denote this fact by a ∈ S

Well-Defined

A set is well-defined, meaning that if S is a set and a is some object, then either a is definitely in S, denoted a ∈ S, or a is definitely not in s, denoted a ∉ S.

Structural Property

A structural property of a binary structure is one that must be shared by any isomorphic structure. Possible Structural Properties: 1. The set has 4 elements 2. The operation is commutative 3. x*x=x ∀ x∈S 4. The equation a*x=b has a solution x ∀a,b∈S Possible NonStructural Properties: 1. The number 4 is an element 2. The operation is called "addition." 3. The elements of S are matrices. 4. S ⊆ C (complex numbers)

Up to isomorphism

All groups with a single element are isomorphic. All grips with just two elements are isomorphic. All groups with just three elements are isomorphic. We use the phrase "up to isomorphism" to express this identification using the equivalence relation ≅ (without lower line). Thus we may say "There is only one group of three elements,up to isomorphism."

Example of Isomorphism

Example of Isomorphism 1. The binary structure <R,+> with operation usual addition is isomorphic to the structure <R+,*> where * is usual multiplication. 1. Define ∅: R →R+ by ∅(x)=e^x for x∈R. (Note e^x>0 ∀x∈R, so ∅(x)∈R+) 2. Assume ∅(x) = ∅(y) ∴ e^x = e^y. ∴ ln e^x = ln e^y ∴ x = y ∅ is one-to-one 3. Assume r∈R+ ∴ ln(r) ∈ R ∴∅ (ln r) = e^ln r = r ∴ ∅ is onto R+ 4. ∀ x, y ∈R, we have ∅(x+y) = e^ (x+y) = e^x * e^y = ∅(x)* ∅(y) ∴ ∅ is an isomorphism ----------------------

Weaker, but similar axioms for a group

1. The binary operation * on G is associative. 2. ∃ a left identity element e in G s.t. e*x =x, ∀x∈G 3. ∀a∈G, ∃ a left inverse a'∈G s.t. a'*a=e If we have a left identity, we can prove a right identity. e*e=e (x'*x)*e = x'*x (x')'*(x'*x)*e = (x')'*(x'*x) [(x')'*x']*x*e = [(x')'*x']*x by assoc [e]*x*e =[e]*x (e*x)*e = x x*e = x is a right identity A left inverse proves a right inverse. a'*a=e (a'*a)*a' = e*a' = a' (a')'*(a'*a)=(a')'*a' a*a'=e ????? See 4.38?

Examples of Groups

4.4 The set Z+ under addition is not a group; no identity element. 4.5 The set of all nonnegative integers (including 0) under addition is still not a group; there is no inverse for 2. 4.6 The familiar additive properties of integers and of rational, real, and complex numbers show that Z, Q, R, and C under addition are abelian groups. 4.7 The set Z+ under multiplication is not a group; no inverse for 3. 4.8 The familiar multiplicative properties of rational, real, and complex numbers show that the sets Q+ and R+ of positive numbers and the sets Q*, R*, and C* of nonzero numbers under multiplication are abelian groups. 4.11 The set M (mxn) (R) of all m x n matrices under matrix addition is a group. the m x n matrix with all entries 0 is the identity matrix. This group is abelian. 4.12 The set M (nxn) (R) of all n x n matrices under matrix multiplication is not a group. The n x n matrix with all entries 0 has no inverse. SO 4.13. the subset S of Mn(R) consisting of all invertible n x n matrices under matrix multiplication is a group.

Abelian

A group G is abelian if its binary operation is commutative.

To show that binary structures are NOT isomorphic:

Example: Different cardinality: The binary structures <Q,+> and <R,+> are not isomorphic because Q has cardinality ℵ0 while |R|≠ℵ0. Note that it is not enough to say that Q is a proper subset of R. Example 3.9 shows that a proper subset with the induced operation can indeed be isomorphic to the entire binary structure. (<Z,+> is isomorphic to <2Z,+> where + is the usual addition) Example: The sets Z and Z+ both have cardinality ℵ0, and there are lots of one-to-one functions mapping Z to Z+. However, the binary structures <Z, *> and <Z+, *> where * is usual multiplication, are NOT isomorphic. In <Z,*>, ∃ 2 elements x s.t. x*x=x, namely 0 and 1. However in <Z+,*>, ∃ only the single element 1. Note: In the event that there are one-to-one mappings of S onto S', we usually show that <S,*> is not isomorphic to <S',*'> by showing that one has some structural property that the other does not possess.

Theorem 4.16 A linear equation in a group has a unique solution

If G is a group with binary operation *, and if a an b are any elements of G, then the linear equations a*x=b and y*a=b have unique solutions x and y in G. (NTS: a'*b is a solution of a*x=b) a*(a'*b)= (a*a')*b = e*b = b (NTS: b*a' is a solution of y*a=b) (b*a')*a = b*(a'*a) = b*e = b Uniqueness: Let y1 and y2 be two solutions. y1*a = b = y2*a ∴y1=y2 by right cancellation (Th 4.15 using a')

Theorem 4.15 Left and Right Cancellation Laws

If G is a group with binary operation *, then the left and right cancellation laws hold in G, that is, a*b=a*c implies b=c, and b*a=c*a implies b=c ∀a,b,c ∈ G. Assume a*b=a*c a'*(a*b) = a'*(a*c) (a'*a)*b=(a'*a)*c by associativity e*b=e*c by inverse b=c by identity

Order

If G is a group, then the order |G| of G is the number of elements in G. Recall that for any set S, |S| is the cardinality of S.

Improper Subgroup Proper Subgroup Trivial Subgroup Nontrivial Subgroup

If G is a group, then the subgroup consisting of G itself is the improper subgroup of G. All other subgroups are proper subgroups. The subgroup {e} is the trivial subgroup of G. All other subgroups are nontrivial.

Subgroup

If a subset H of a group G is closed under the binary operation of G and if H with the induced operation from G is itself a group, then H is a subgroup of G. We shall let H≤G or G≥H denote that H is a subgroup of G, and H<G or G>H shall mean H≤G but H≠G. Thus <Z,+> < <R,+> BUT <Q+, mult> is not a subgroup of <R,+> even though Q+ ⊂ R. Every group G has as subgroups G itself, and {e} where e is the identity element of G. So in this case the identity element of Q+ under multiplication is 1, but the identity element of R under addition is 0. Ex. Q+ under multiplication is a proper subgroup of R+ under multiplication. The nth roots of unity in C form a subgroup Un of the group C* of nonzero complex numbers under multiplication

Theorem 4.17 Identity Elements and Inverses are Unique

In a group G with binary operation *, there is only one element e in G such that e*x = x*e = x, ∀ x ∈G Likewise, there is only one element a' in G such that a'*a = a*a=e

Closed Under Induced Operation

Let * be a binary operation on S and let H be a subset of S. The subset H is closed under * if for all a,b ∈ H we also have a*b ∈ H. In this case, the binary operation on H given by restricting * to H is the induced operation of * on H.

Isomorphism

Let <S, *> and <S', *'> be binary algebraic structures. An isomorphism of S with S' is a one-to-one function ∅ mapping S onto S' such that ∅(x * y) = ∅(x) *' ∅(y) for all x, y ∈ S. (homomorphism property) If such a map ∅ exists, then S and S' are isomorphic binary structures, which we denote by S≅S' (Only include ONE lower straight line), omitting the * and *' from the notation.

Identity Element

Let <S, *> be a binary structure. An element e of S is an identity element for * if e*s = s*e = s for all s ∈ S.

Corollary 4.18

Let G be a group. ∀ a,b ∈ G, (a*b)' = b'*a' (a*b)*b'*a' =a*(b*b')*a'=a*e*a'=(a*e)*a'=a*a'=e ∴ ∀ a,b ∈G, the inverse of (a*b) or (a*b)' = b'*a'

Note: U

Note: U = {z ∈ C | |z| = 1}, so that U is the circle in the Euclidean plane with center at the origin and radius 1. Reminder: |z1 z2| = |z1| |z2| shows that the product of two numbers in U is again the number U; we say that U is closed under multiplication.

Injection

One-to-One Let ∅ be a function on S. Suppose then ∅(x) = ∅(y) in S' and deduce that x=y in S.)

Bijection

One-to-one (injection) AND Onto (surjection)

Surjection

Onto Let ∅ be a function on S. Suppose that s' ∈ S' is given and show that there does exist s ∈ S such that ∅(s) = s'.

General Linear Group of degree n

The group of invertible n x n matrices is of fundamental importance in linear algebra. Usually denoted GL(n,R) Recall 4.13. the subset S of Mn(R) consisting of all invertible n x n matrices under matrix multiplication is a group.

Klein 4-group

There are two different types of group structures of order 4. The group V is the Klein 4-group,and the notation V comes from the German word Vier for four. The group Z4 is isomorphic to the group U4 = {1,i,-1,-i} of fourth roots of unity under multiplication. The only nontrivial subgroup of Z4 is {0,2}. Note {0,3} is not a subgroup b/c 3+3 = 2 ∉ {0,3} However, V has three nontrivial, proper subgroups: {e,a}, {e,b}, {e,c} Note {e,a,b} is not a subgroup b/c ab=c ∉{e,a,b}

Empty Set

There is exactly one set with no elements. It is the empty set and is denoted by ∅.