Homomorphisms, Isomorphisms, and Automorphisms

4 Functions between Algebraic Structures One rule with almost no exceptions is that mathemat- ical structures are not studied in isolation: as well as the structures themselves one looks at certainfunctions defined on those structures. In this section we shall see which functions are worth considering, and why. (For a discussion of functions in general, seethe language and grammar of mathematics[I.2 §2.2].)

4.1 Homomorphisms, Isomorphisms, and

alluded to in our discussion of groups. An automor- phism ofXis a function fromXto itself that preserves the structure (which now comes in the form of state- ments likeab=c). The composition of two automor- phisms is clearly a third, and as a result the automor- phisms of a structureX form a group. Although the individual automorphisms may not be of much inter- est, the group certainly is, as it often encapsulates what one really wants to know about a structureXthat is too complicated to analyze directly.

A spectacular example of this is when X is a field.

To illustrate, let us take the example of_Q(√

2). Iff : Q(√

2)→ Q(√

2)is an automorphism, then f (1)=1.

(This follows easily from the fact that 1 is the only mul- tiplicative identity.) It follows thatf (2)=f (1+1)= f (1)+f (1) = 1+1 = 2. Continuing like this, we can show thatf (n)=nfor every positive integern.

Thenf (n)+f (−n) = f (n+(−n)) = f (0) = 0, so f (−n)= −f (n)= −n. Finally,f (p/q)=f (p)/f (q)= p/qwhenpandqare integers withq=0. Sof takes every rational number to itself. What can we say about f (√

2)? Well,f (√ 2)f (√

2)=f (√ 2·√

2)=f (2)=2, but this implies only thatf (√

2)is√ 2 or−√

2. It turns out that both choices are possible: one automorphism is the “trivial” one,f (a+b√

2)=a+b√

2, and the other is the more interesting one,f (a+b√

2) = a−b√ 2.

This observation demonstrates that there is no alge- braic difference between the two square roots; in this sense, the field_Q(√

2)does not know which square root of 2 is positive and which negative. These two automor- phisms form a group, which is isomorphic to the group consisting of the elements±1 under multiplication, or the group of integers modulo 2, or the group of sym- metries of an isosceles triangle that is not equilateral, or. . . . The list is endless.

The automorphism groups associated with certain field extensions are calledGalois groups, and are a vital component of the proof ofthe insolubility of the quintic [V.21], as well as large parts of algebraic number theory[IV.1].

An important concept associated with a homomor- phismφbetween algebraic structures is that of aker- nel. This is defined to be the set of all elementsxofX such thatφ(x)is the identity element ofY(where this means the additive identity ifXandY are structures that involve both additive and multiplicative binary operations). The kernel of a homomorphism tends to be a substructure ofXwith interesting properties. For instance, ifGandK are groups, then the kernel of a homomorphism fromGtoKis a normal subgroup of

G; and conversely, ifHis a normal subgroup ofG, then thequotient map, which takes each elementg to the left cosetgH, is a homomorphism fromGto the quo- tient groupG/Hwith kernelH. Similarly, the kernel of any ring homomorphism is anideal[III.81], and every ideal I in a ringR is the kernel of a “quotient map”

fromRtoR/I. (This quotient construction is discussed in more detail inrings, ideals, and modules[III.81].) 4.2 Linear Maps and Matrices

Homomorphisms between vector spaces have a dis- tinctive geometrical property: they send straight lines to straight lines. For this reason they are called lin- ear maps, as was mentioned in the previous subsec- tion. From a more algebraic point of view, the struc- ture that linear maps preserve is that of linear combi- nations: a functionf from one vector space to another is a linear map iff (au+bv) = af (u)+bf (v)for every pair of vectorsu,v∈Vand every pair of scalars aandb. From this one can deduce the more general assertion thatf (a1v1+ · · · +anvn)is always equal to a1f (v1)+ · · · +anf (vn).

Suppose that we wish to define a linear map fromV toW. How much information do we need to provide? In order to see what sort of answer is required, let us begin with a similar but slightly easier question: how much information is needed to specify a point in space? The answer is that, once one has devised a sensible coordi- nate system, three numbers will suffice. If the point is not too far from Earth’s surface then one might wish to use its latitude, its longitude, and its height above sea level, for instance. Can a linear map fromV toW similarly be specified by just a few numbers?

The answer is that it can, at least ifVandWare finite dimensional. Suppose thatVhas a basisv₁, . . . ,v_n, that Whas a basisw1, . . . ,w_m, and thatf :V →Wis the lin- ear map we would like to specify. Since every vector in Vcan be written in the forma₁v₁+· · ·+a_nv_nand since f (a1v₁+· · ·+a_nv_n)is always equal toa₁f (v₁)+· · ·+

anf (vn), once we decide whatf (v1), . . . , f (vn)are we have specifiedf completely. But each vectorf (v_j)is a linear combination of the basis vectorsw1, . . . ,wm: that is, it can be written in the form

f (vj)=a1jw1+ · · · +amjwm.

Thus, to specify an individualf (vj)needsmnumbers, the scalarsa_1j, . . . , a_mj. Since there arendifferent vec- torsvj, the linear map is determined by themnnum- bersaij, whereiruns from 1 tomandjfrom 1 ton.

These numbers can be written in an array, as follows:

⎛

⎜⎜

⎜⎜

⎜⎝

a11 a12 · · · a1n

a21 a22 · · · a2n

... ... . .. ... am1 am2 · · · amn

⎞

⎟⎟

⎟⎟

⎟⎠.

An array like this is called amatrix. It is important to note that a different choice of basis vectors forV and W would lead to a different matrix, so one often talks of the matrix off relative to a given pair of bases (a basis forVand a basis forW).

Now suppose thatfis a linear map fromVtoWand thatgis a linear map fromUtoV. Thenf gstands for the linear map fromUtoW obtained by doing firstg, thenf. If the matrices of f andg, relative to certain bases ofU,V, andW, areAandB, then what is the matrix off g? To work it out, one takes a basis vectoru_k ofUand applies to it the functiong, obtaining a linear combinationb_1kv1+· · ·+bnkvnof the basis vectors of V. To this linear combination one applies the function f, obtaining a rather complicated linear combination of linear combinations of the basis vectorsw1, . . . ,wm

ofW.

Pursuing this idea, one can calculate that the entry in rowiand columnjof the matrixP off gisa_i1b_1j+ a_i2b_2j+· · ·+a_inb_nj. This matrixPis called theproduct ofAandBand is writtenAB. If you have not seen this definition then you will find it hard to grasp, but the main point to remember is that there is a way of calcu- lating the matrix forf gfrom the matricesAandBoff andg, and that this matrix is denotedAB. Matrix mul- tiplication of this kind is associative but not commuta- tive. That is,A(BC)is always equal to(AB)CbutABis not necessarily the same asBA. The associativity fol- lows from the fact that composition of the underlying linear maps is associative: ifA,B, andCare the matrices off,g, andh, respectively, thenA(BC)is the matrix of the linear map “doh-then-g, thenf” and(AB)C is the matrix of the linear map “doh, theng-then-f,” and these are the same linear map.

Let us now confine our attention toautomorphisms from a vector spaceV to itself. These are linear maps f:V→V that can be inverted; that is, for which there exists a linear map g : V → V such that f g(v) = gf (v)=vfor every vectorvinV. These we can think of as “symmetries” of the vector spaceV, and as such they form a group under composition. IfVisndimen- sional and the scalars come from the field_F, then this group is called GLn(_F). The letters “G” and “L” stand for

“general” and “linear”; some of the most important and

difficult problems in mathematics arise when one tries to understand the structure of the general linear groups (and related groups) for certain interesting fields_F(see representation theory[IV.9 §§5,6]).

While matrices are very useful, many interesting linear maps are between infinite-dimensional vector spaces, and we close this section with two examples for the reader who is familiar with elementary calcu- lus. (There will be a brief discussion of calculus later in this article.) For the first, letV be the set of all func- tions from_Rto_Rthat can be differentiated and letW be the set ofall functions from_Rto_R. These can be made into vector spaces in a simple way: iff andg are functions, then their sum is the functionhdefined by the formulah(x)=f (x)+g(x), and ifais a real number thenaf is the functionkdefined by the for- mulak(x)=af (x). (So, for example, we could regard the polynomialx²+3x+2 as a linear combination of the functionsx²,x, and the constant function 1.) Then differentiation is a linear map (fromV toW), since the derivative(af+bg) isaf +bg. This is clearer if we write Dffor the derivative off: then we are saying that D(af+bg)=aDf+bDg.

A second example uses integration. LetVbe another vector space of functions, and letube a function oftwo variables. (The functions involved have to have certain properties for the definition to work, but let us ignore the technicalities.) Then we can define a linear mapT on the spaceV by the formula

(T f )(x)=

u(x, y)f (y)dy.

Definitions like this one can be hard to take in, because they involve holding in one’s mind three different lev- els of complexity. At the bottom we have real numbers, denoted byxandy. In the middle are functions likef, u, andT f, which turn real numbers (or pairs of them) into real numbers. At the top is another function,T, but the “objects” that it transforms are themselves func- tions: it turns a function likef into a different func- tionT f. This is just one example where it is important to think of a function as a single, elementary “thing”

rather than as a process of transformation. (See the dis- cussion of functions inthe language and grammar of mathematics[I.2 §2.2].) Another remark that may help to clarify the definition is that there is a very close analogy between the role of the two-variable function u(x, y)and the role of a matrixa_ij (which can itself be thought of as a function of the two integer vari- ables iandj). Functions likeuare sometimes called kernels(which should not be confused with kernels of

homomorphisms). For more about linear maps between infinite-dimensional spaces, seeoperator algebras [IV.15] andlinear operators[III.50].