Stages of Algebra:

Stages of Algebra:

Algebraic Expression: Rhetorical Algebra, Syncopated Algebra and Symbolic Algebra.

Conceptual Stages: Geometric Stages, Static equation solving stage, Dynamic functional stage, Abstract Stage.

Liberation of Algebra: Role of Hamiltonian Quaternion (1,i,j,k), Grassmann hypercomplex numbers x₁e₁+x₂e₂+…+x_ne_n, Caley Matrix algebra. (See History of mathematics for detail)                                                                

a.      Structures of modern algebra and their teaching

Twentieth century mathematics is characterized by its emphasis on the systematic investigation of a number of abstract mathematic structures. Many such basic structures come under Abstract Algebra. The study of groups, rings, fields is not only the students first introduction to modern mathematical thinking but provides some of the essential crutches used by mathematicians morning and evening. What constitute abstract structures in mathematics? Crudely speaking, these may be viewed as decorations on a set eg. A topological space is a set with a prescribed collection of subsets to be called open sets satisfying some specific axioms. Axioms are considered as ‘self -evident truths’. Today many textbooks avoid the word ‘axiom’ with ‘properties’.

For example: In the defining the group you can write: A group (G,*) is a set G together with a binary operation * on G, such that the following properties (axioms) are satisfied:…..

 The introduction of abstract mathematical structures in the curricula in Nepal was late by several decades when compared to many advanced countries. During the last generation specially at the advanced level were introduced to this effect in several stages.

 

Modern algebra can be classified into abstract and linear algebra.

Abstract Algebra:

What Is Abstract Algebra?

 

School algebra can be seen as a generalization of arithmetic in which the variables are numbers and the expressions and equations are formed with the four arithmetic operations.  Abstract algebra is a generalization of school algebra in which the variables can represent various mathematical objects, including numbers, vectors, matrices, functions, transformations, and permutations, and in which the expressions and equations are formed through operations that make sense for the particular objects: addition and  multiplication for matrices, composition for functions, and so on.  This section provides a short sketch of abstract algebra in order to highlight ideas of structure and to present the terms, concepts, notations, and perspectives that undergird the research questions and subsequent analysis.

Abstract algebra consists of axiomatic theories that provide opportunities to consider many different mathematical systems as being special cases of the same abstract structure.  The theories are called axiomatic because the structures are defined by axioms. 

Group theory is “one of the oldest (and also one of the simplest) of axiomatic theories” (Bourbaki, 1950, p. 224).  

Consider, for example, the following four mathematical systems: 

1. The integers {… , -3, -2, -1, 0, 1, 2, 3, …} under the operation of addition.  This

system is denoted Z.

2. The whole numbers less than a given whole number n, {0, 1, 2, … , n – 1}, under the operation of addition, where addition is given by the remainder after dividing the usual sum by n.  This system is denoted Z_n.

3. The translations of the plane, where the operation is given by composition, that is, following one translation by another.

4. The set of 2 2 matrices of real numbers with determinant 1, under matrix

multiplication.

Each of these examples consists of a set of elements (numbers or translations) together  with an operation that specifies how to combine two of the elements to get an element that is also in the set.  Because the operation combines two elements, it is often called a binary operation.  In order to talk about these examples simultaneously, the operation is denoted by *, where the interpretations are addition, addition “modulo n,” composition,

and matrix multiplication, respectively, in the four examples.  

With some work, it is possible to see that each of these systems satisfies the following axioms:

1. Associativity.  For any three elements, x, y, and z, in the set (not necessarily distinct),

(x*y)*z = x*(y*z).  

2. Identity.  There is an element, e, in the set, such that for any x in the set,

 e*x = x =x*e.  (For addition of integers, the identity is 0; for addition modulo n, the identity is 0; for translations of the plane, it is the “identity” translation that leaves every point fixed; for matrices under multiplication, it is the “identity” matrix with 1s on the diagonal and 0s elsewhere.)

3. Inverse.  For each element x in the set, there is an element y in the set such that x*y = e = y*x.  

A fourth (or zero) axiom, closure, is built into the requirements of a binary operation: that the combination of two elements gives an element that still lies in the set.  It should be pointed out that commutativity is not one of the axioms, and it is not hard to see that  matrix multiplication is not commutative.

Any set and operation that together satisfy these axioms is said to be a group.  When the operation is also commutative, the group is said to be Abelian.  The advantage of the axiomatic approach is that any result (i.e., theorem) that can be proved on the basis of the axioms alone necessarily applies to all four examples and also to any other mathematical system that satisfies the axioms.  

The important results in group theory depend upon a collection of related concepts.  A subgroup, for example, is a subset of a group, which is itself a group under the group’s operation.  The role of structure again returns to the fore with the concept of isomorphism.  On a high level, the group axioms define an algebraic structure that applies to a broad collection of mathematical systems.  The axioms create the rudimentary structure to which all groups must conform.  At a lower level, every specific group is a mathematical system with its own internal structure.  An important abstraction can occur when two groups appear in different settings and yet are “essentially the same.”  The intuitive idea is that two groups are structurally the same, or isomorphic, if they differ only in the names of their elements and operation.  Demonstrating that two groups are isomorphic requires finding a renaming that preserves the group operation.  Such a renaming, which is essentially a function that takes elements from one group to the other, is called an isomorphism.

It should be pointed out that the above mathematical systems and other standard examples may not be familiar to undergraduates in a first course in abstract algebra.  Thus, some of the student’s energy must be spent trying to build some familiarity with the examples. 

Taken together, these examples and the concepts of group, subgroup, and isomorphism constitute the fundamental concepts of group theory for the purpose of this study.

I distinguish as “advanced concepts of group theory” those concepts that require the construction of new objects.  Given a subgroup H, one can create a left coset of the subgroup by multiplying an element a of the group on the left by each of the elements in the subgroup.  The coset is denoted aH.  When the set of left cosets forms a group by extending the group operation to the cosets, the resulting group is called a quotient group, and the subgroup that gave rise to the cosets is said to be normal.

Other important mathematical structures are rings and fields.  In ring theory, there are two operations, typically called multiplication and addition.  Examples are the arithmetic of integers, of matrices, and of polynomials in one variable with integer coefficients.  A field is essentially a ring in which multiplication is commutative and division is also possible, except, of course, division by zero.  Examples are the rational numbers, the

complex numbers, and the integers modulo p, where p is prime.

The Big Ideas of Abstract Algebra

A course in abstract algebra is the place where students might extract common features from the many mathematical systems that they have used in previous mathematics courses, such as calculus, linear algebra, and school algebra.  Students have opportunities to develop deeper understandings of concepts such as identity, inverse, equivalence, and function.  What is shared, for example, by the identity for multiplication of real numbers, the identity matrix, and the identity function?  What is the common idea behind the

inverse of a function, the inverse of a matrix, and the multiplicative inverse of a number? 

In abstract algebra, students can also learn about the importance of precise language in mathematics and about the role of definitions in supporting such precision.  Mathematics is also about noticing when things are the same and being able to describe how they are different.  In abstract algebra, this naïve notion of “sameness” becomes formalized in the concept of isomorphism.

Thus, it is clear that the concepts in abstract algebra provide guiding themes, principles, and sensibilities that pervade mathematics.  It is not so clear, however, what sequence of topics from abstract algebra can be constructed to help students recognize and appreciate such themes.  And, in particular, it is not clear whether an abstract algebra course intended for mathematics majors, as it is typically taught, can serve such a role. 

When the population of students in an abstract algebra course includes future teachers (which may be almost always), these big ideas, such as inverse and identity, are particularly important because they can help teachers connect advanced mathematics with high school mathematics in ways that can strengthen and deepen their understandings of the mathematics they will teach.  Of course, it is also crucial that future teachers are able to employ those new understandings in their teaching, but that concern takes us beyond the scope of this study.  

Sfard (1995) gives a detailed a description of the historical development of algebra with strong connections to the teaching and learning of both school and abstract algebra, providing compelling support for the claim that historical-critical and psychogenetic studies should converge (Piaget & Garcia, 1989, p. 108).  According to Sfard, group theory arose out of the work of Lagrange and Ruffini, who noticed that methods of solving polynomial equations depended on permutations of the roots.  Soon permutations and then, with Cauchy, operations on those permutations became objects of attention. 

Galois defined the notion of a group by declaring interest in the structure imposed on the permutations by the so-called substitutions.  Cayley freed the concept from any commitment as to the nature of the elements, focusing instead on the manipulations. 

With the invention of the concept of group, the seeds had been planted for algebra to become a science of abstract structures.

Kleiner (1986) describes four lines of inquiry that coalesced toward the end of the nineteenth century to form the area we now call abstract algebra.  First, the techniques from classical algebra for solving polynomial equations led to the permutation groups. 

Second, questions in number theory led to the finite Abelian groups.  Third, attempts to unify and organize geometry led to transformation groups.  Finally, roots in analysis led to investigation of continuous transformation groups.  One response to this account is to use historically important problems to provide pedagogical and intellectual motivation in the teaching of abstract algebra (see Kleiner, 1995).

Nicholson’s (1993) account of the slow historical development of the concept of quotient group can provide additional sources for cognitive roots to be exploited.  She suggests several obstacles that were overcome by the mathematics community during the development of this concept.  First, the community needed an abstract concept of group that was not dependent on any particular representation.  Second, the community needed the concept of equivalence (modulo a subgroup).  Finally (and most importantly), the community needed to realize that the elements of the quotient group are not like the elements of original group, but are equivalence classes—sets.  All of these historical developments provide clues about what might be the issues for students learning the subject

 

Group Theory: The notion of group did not simply spring into existence, however, but is rather the culmination of a long period of mathematical investigation. The first formal definition of an abstract group in the form in which we use it appearing in 1882. The definition of an abstract group has its origins in extremely old problems in algebraic equations, number theory and arose because very similar techniques were found to be applicable in a variety of situations.

We illustrate with a few of the disparate situations in which the ideas later formalized into the notion of an abstract group were used:

1.      In number theory the very object of study, the set of integers, is an example of a group.  Consider for example what we refer to as “Euler’s Theorem” :  for every integer ‘a’ relatively prime to ‘n’.

This was proved in 1761 by Euler using “group theoretic” ideas of Lagrange, long before the first formal definition of a group.

2.      Investigations into the question of rational solutions to algebraic equations of the form y²=x³-2x there are infinitely many points such as (0,0), (-1,1), (2,2), (9/4,-21/8)… . today the curve above is referred to as an “elliptic curve” and these questions are viewed as two different aspects of the same thing- the fact that this geometric operation on points can be used to give the set of points on an elliptic curve the structure of a group.

3.      By 1824 it was known that there are formulas giving the roots of quadratic, cubic and quartic equations. In 1824 Abel prove that such a formula for the roots of a quintic is impossible [the general equation of degree n cannot be solved by radicals for n>4]. The proof of this theorem is based on the idea of examining what happens when the roots are permuted amongst themselves. The collection of such permutations has the structure of a group ‘permutation group’. This idea culminated in the beautiful work of Evarste Galois in 1830-32 working with explicit group of substitutions. He was the first person who used the word “group”. Today this work is referred to as Galois Theory. Similar explicit groups were being used in geometry as transformation group by Cayley around 1850, Jordan by 1867, Klein by 1870.  

Concepts contained in group theory 

Binary Operation:

     Modular Algebra:

-      Concept of equivalence relation: The relation a b(mod n) is the equivalence relation. How?

Different examples of group: dihedral group, symmetric group, matrix group, quaternion group etc.

Subgroups:

Definition and examples. eg: Intersection of two subgroup of any group is subgroup but union may not be. For eg: ({0,6},+₁₂) and ({0,4,8},+₁₂) are two subgroup of group (z,+₁₂) of integer modulo 12 but what about the union.

Centre, Centerlizer and Normalizer: C_G(H)<N_G(H)<G

Cyclic Groups and Cyclic subgroups, normal subgroups etc.

Quotient Groups and Homomorphism

           Definition and examples

Cosets and Lagrange’s Theorem

Isomorphism theorems

Composition Series and the Holder program

Transposition and  the alternating group

Group Actions

Direct product and Abelian group

Nilpotent and solvable group.

 

Ring Theory:

The theory of groups is concerned with general properties of certain objects having an algebraic structure defined by a single binary operation. The study of rings is concerned with objects possessing two binary operations (addition and multiplication) related by the distributive laws. Some concepts of ring theory are analogues for the basic points of development in the structure theory of groups. In particular, subrings, quotient rings , ideals (analogues with normal subgroup) and ring homomorphism. The general theory of ring is arise naturally from the presence of two binary operations. Questions concerning multiplicative inverse leads to the notion of fields and eventually to the construction of some specific fields such as finite fields. The study of the arithmetic (divisibility, GCD etc) of rings leads to the notion of primes and unique factorizations and are then applied to rings of polynomials.

Structure of content:

1.      Basic definition and examples:

2.      Polynomial rings, matrix rings, and group rings

3.      Ring homomorphisms and quotient rings

4.      Ideals and properties

5.      Euclidean domain, PID and UFD

6.      Polynomial ring (simple concepts)

Module Theory:

The use of modules was pioneered by one of the most prominent mathematicians Emmy Nother, who led the way in demonstrating the power and elegance of this structure. The vector spaces are just special types of modules which arise when the underlying ring is a field. If R is a ring the definition of an R-module M is closely analogous to the definition of a group action where R plays the role of the group and M the role of the set.

1.      Basic definitions and Examples

2.      Quotient Modules and module homomorphism

3.      Generation of modules, Direct Sums, and free Modules

4.      Tensor Product of Modules

5.      Exact Sequences-Projective, injective and flat module.

Vector Spaces

1.      Definition and Basic Theory

2.      The matrix of a linear transformation

3.      Dual vector spaces

 

Field Theory:

1.      Basic theory of field etensions

2.      Algebraic extensions

3.      Spliting fields

Definitions:

Groupoid, Semigroups, Monoids, Groups and Abelian Groups:

1.      A nonempty set G together with binary operation * is said to be a group (G, *) if the following postulates (axioms/properties) holds under the binary operation * in G.

i.          G is closed under *, ie.  (Closure axiom)Obvious on binary operation.

ii.         G is associative under *, i.e   (Associative)

iii.        There exists an element e in G, called an identity of G such that for all a  we have a*e=e*a=a.

iv.        For each a  there is an element , called an inverse of a, such that a*

2.   The group (G,*) is called abelian (or commutative) if a*b=b*a for all a,b

 

i	i,ii	i,ii,iii	i-iv	1,2
Groupoid (Quasi group)	Semigroup	Monoid	Group	Abelian Group

 

Rings:

Definition

1) A ring R is a set together with two binary operations + and x (called addition

and multiplication) satisfying the following axioms:

(i)                 (R. +) is an abelian group,

(ii)        x is associative: (a x b) x c = a x (b x c) for all a, b, c  R,

(iii)       the distributive laws hold in R : for all a, b, c  R

(a+b)xc=(axc)+(bxc) and ax(b+c)=(axb)+(axc).

(2) The ring R is commutative if multiplication is commutative.

(3) The ring R is said to have an identity (or contain a I) if there is an element I  R  with lxa=axl=a for all a  R.

 

Definition. A ring R with identity 1, where 1  0 is called a division ring (or skew

field) if every nonzero element a  R has a multiplicative inverse, i.e., there exists

b  R such that ab = ba = I. A commutative division ring is called a field.

Module Theory:

Definition: Let R be a ring (not necessarily commutative nor with I). A left R -module

or a left module over R is a set M together with

1.      a binary operation + on M under which M is an abelian group, and

2.   an action of R on M (that is, a map R x M  M) denoted by rm, for all r  R

and for all m  M which satisfies

i.          (r + s)m = rm + sm, for all r, s  R, m  M,

ii.          (rs)m = r(sm), for all r, s  R, m  M, and

iii.         r(m + n) = rm + rn, for all r  R, m, n  M.

If the ring R has a 1 we impose the additional axiom:

iv.        1m = m, for all m  M.

Vector Space

Let F be a field. A vector space over F consists of an abelian group V under addition together with an operation of scalar multiplication of each element of V by each element of F on the left s.t

i.          av

ii.                  a(bv)=(ab)v

iii.                (a+b)v=(av)+(bv)

iv.                a(u+v)=au+av

v.                  1v=v.

 

Ring	Field	Module	Vector Space
·      Two binary operations + and	Two binary operations + and	Two binary operations + and	Two binary operations + and
·      (R,+) is abelian group, (R,  is a semigroup with distributive property.	(F,+) is an abelian group and ( ) is abelian group.	(M,+) is an abelian group and scalar multiplication (rm  is closed and distributive over scalar.	(V,+) is an abelian group, v  and scalar multiplication is closed and distributive, 1v=v.
·      Every ring is an R-module	F is a F-module, F is vector space over F.	If R is a division Ring then unitary R-module is called vector space.	V is an F module.
·		Module is a generalization of vector space.
·
·
·

 

 

Subgourp:

A non-empty subset H of G is said to be subgroup of G if H itself form a group under same binary operation as in G.

For example:

1.      Set of integer is the subgroup of the set of Real number under addition binary operation.

2.      The set A={(1), (123),(132)} is subgroup of S₃ under permutation multiplication.

Some important results in subgroup:

1.   A nonempty subset H of group (G,*)is a subgroup iff ab^-1

2.   A non empty finite subset H of group G is a subgroup of G iff ab , a,b

3.      The intersection of two subgroup of a group is also a subgroup. But Union may not be subgroup.

Sub-Rings:

A nonempty subset S of a ring R is said to be a sub ring of R if S itself forms a ring under the same binary operations of R.

For example:

1.      The ring (Z,+,*) is a subring of (Q,+,*)

2.   The set 2Z={2n:n  is a sub ring of (Z,+,*).

Note:

1.   A nonempty subset S of a ring R is a sub ring of R if for each a,b  i) a-b  ii) a*b

2.      The intersection of two subrings of a ring is also a subring.

Normal Subgroup:

A subgroup H of a group G is a normal (invariant) subgroup of G if g^-1Hg=H for all g .

Thus normal subgroups are precisely those important subgroups of a group having the property that for the cosets(left and right are the same), the induced operation is well defined and the cosets form a group.

Ideal: (Analogue of the definition of a normal subgroup)

An additive subgroup (N,+) of a ring R satisfying  rN  and Nr  for all r  is an ideal (or two sided ideal)  of R.

Every ideal I is the subring of R but every subring may not be ideal of R

For example: Let Z be the set of integers and I=2Z is the set of even integer which is an ideal of R.

But Z is the subring of Q. But Z is not ideal of Q. For let, 3/5 , 2  but 3/5.2 . So Z is not ideal of Q.

Factor/Quotient Group:

Quotient Ring: (Analogue of the definition of the factor group): If N is an ideal in a ring R, then the ring of cosets  r+N under the induced operations is the quotient ring, or factor ring or residue class ring of R modulo N and is denoted by R/N.

Group Homomorphism:

Let (G,*) and (G’,*) be two group. The mapping  is said to be group homo if

For Example: The mapping  Is the group homo but the mapping  is not the group homo.

Ring Homomorphism:

Let (R,+,*) and (R’,+,*) be two rings, a mapping  is said to be ring homo if  a) f(a+b)=f(a)+f(b) and f(a*b)=f(a)*f(b)

Module Homomorphism:

Let M and N are any two R-module. The function  is said to be R-module homomorphism if f(a+b)=f(a)+f(b), f(ra)=rf(a)  a,b

Show by example every group homo is module homo.

Also module homo may not be ring homo and vice versa.

Let M and N be two arbitrary abelian groups and  Z be the Z module.

Let  be group homomorphism. Let a,b . We have f(a+b)=f(a)+f(b)

Let  then

            f(na)=f(a+a+… to n times)

                        = f(a)+f(a)+… to n times

                        =n f(a).             f(na)=nf(a).

Hence, each group homo is an Z-module Homo.

In the ring of integer Z, the function defined as  by f(z)=2z, z

f is module homo:

 Since f(m+n)=f(m)+f(n)

            And f(rm)=f(m+m+…+r times)

                            = f(m)+f(m)+…+f(m)  r times= rf(m).

Hence f is module homo.

Bur for ring homo: Let f is module homo then f(m+n)=f(m)+f(n) and

                                    f(rm)=2rm=r.2m f(r).f(m). So f is not ring homo.

Let us consider the ring of complex number C. Defined  by f(x)=

F is ring homo:

1.   f(x+y)=  = =f(x)+f(y).

2.   f(xy)= =f(x).f(y). So f is ring homo.

But for module homo:

1.   f(x+y)=  = =f(x)+f(y).

2.   f(i.i) . f(i.i)=f(-1)=-1 and if(i)=1.

Hence f is not module homo.

Fundamental theorem of group homo, ring homo and module homo.

-          Let  be onto group homo. Then G/ker

-     Let  be ring homo with R onto R’ with kernel N then R/N

-     Let  is an onto module homo with kernel K then M/K

Compare and contrast these theorems.          

 

Difficulties in learning Algebra:

In response to the difficulties students usually have with Lagrange’s theorem, Johnson

(1983) notes that the traditional proof involves cosets and equivalence relations, both of which are new concepts to most students.  As Lagrange’s theorem is usually used to prove the more intuitive theorem that the order of an element divides the order of the group, Johnson suggests proving the latter result first, for it follows quite naturally from the decomposition of a permutation into disjoint cycles.  Of course, this approach assumes the students are familiar with permutation groups, and such an assumption might be unwarranted.  Holton and Wenzel (1993) describe an abstract algebra course in which Lagrange’s theorem is preceded by cooperative learning via examples.  Rejecting the traditional approach of “exposition, exhortation and regurgitation” (p. 883), they found that students were able to conjecture the theorem and many of the necessary lemmas.  Although it was not a formal research study, the description of the classroom environment was compelling.  

Groups and Binary Operations

“Is Z₃ a subgroup of Z₆?”  The short answer to this question is no because the operations in the two groups are different.  More specifically, because Z₃ is the set {0, 1, 2} under addition modulo 3, and Z₆ is the set {0, 1, 2, 3, 4, 5} under addition modulo 6, the operations are not the same.  For example, 2 + 2 is 4 in Z₆   But 1 in Z_3.Nonetheless, the subset{0,2,4} of Z₆ is simultaneously a subgroup of Z₆ and isomorphic to Z₃. From this sense Z₃ is subgroup of Z₆.Two reasons for students’ difficulties with this question.  First, although students think of a group as a set, they are not always sufficiently aware of the operation (Dubinsky et al., 1994).  Some students use a powerful result inappropriately, saying that Z₃ is a subgroup of Z₆ by Lagrange’s theorem because 3 divides 6 (Hazzan & Leron, 1996). 

Just as a group is a set with structure provided by an operation, a subgroup is not merely a subset of a group but rather a substructure, and the structure is provided by the operation of the group.  General insight into the structure can be provided by Lagrange’s theorem, which says that in a finite group the order of a subgroup (the number of elements in the subgroup) must be a factor of the order of the group.  The converse of the theorem is false in general.

 

Linear Algebra (Student Learning Opportunities)

i.                    Linear Maps and Matrices

ii.                  Bilinear Forms and Standard Operators

iii.                Algebraic Properties of Linear Transformation

iv.                Spectral Theorem and Primary decomposition theorem

 

b.      Prepare any three lessons for teaching from algebra of bachelor level using different strategies at least one of them must be in PBL model.