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CHAPTER 1
INTRODUCTION
1.1. Algebraic Laws.
A large part of algebra is concerned with systems of elements which, like numbers, may be combined by addition or multiplication or both. We are given a system whose elements are designated by letters a, b, c, .... We write S = S(a, b, c, ...) for this system. The properties of these systems depend upon which of the following basic laws hold:
Closure Laws. A0. Addition is well defined. M0. Multiplication is well defined.
These mean that, for every ordered pair of elements, a, b of S, a + b = c exists and is a unique element of S, and that also ab = d exists and is a unique element of S.
Associative Laws. A1. (a + b) + c = a + (b + c) M1. (ab)c = a(bc)
Commutative Laws. A2. b + a = a + b M2. ba = ab
Zero and Unit. A3. 0 exists such that 0 + a = a + 0 = a for all a. M3. 1 exists such that 1a = a1 = a for all a.
Negatives and Inverses. A4. For every a, -a exists such that (-a) + a = a + (-a) = 0.M4.* For every a ≠ 0, a-1 exists such that (a-1) a = a(a-1) = 1.
Distributive Laws. D1. a(b + c) = ab + ac. D2. (b + c) a = ba + ca.
Definition: A system satisfying all these laws is called a field. A system satisfying A0, -1, -2, -3, -4, M0, -1, and D1, -2 is called a ring.
It should be noted that A0–A4 are exactly parallel to M0–M4: except for the nonexistence of the inverse of 0 in M4. In the distributive laws, however, addition and multiplication behave quite differently. This parallelism between addition and multiplication is exploited in the use of logarithms, where the basic correspondence between them is given by the law:
log (xy) = log x + log y.
In general an n-ary operation in a set S is a function f = f(a1 ..., an) of n arguments (a1 ..., an) which are elements of S and whose value f(a1 ..., an) = b is a unique element of S when f is defined for these arguments. If, for every choice of a1 ..., an in S, f(a1 ..., an) is defined, we say that the operation f is well defined or that the set S is closed with respect to the operation f.
In a field F, addition and multiplication are well-defined binary operations, while the inverse operation f(a) = a-1 is a unary operation defined for every element except zero.
1.2. Mappings,
A very fundamental concept of modern mathematics is that of a mapping of a set S into a set T.
Definition: A mapping a of a set S into a set T is a rule which as signs to each x of the set S a unique y of the set T. Symbolically we write this in either of the notations:
α: x [right arrow] y or y = (x)α.
The element y is called the image of x under a. If every y of the set T is the image of at least one x in S, we say that a is a mapping of S onto T.
The mappings of a set into (or onto) itself are of particular importance. For example a rotation in a plane may be regarded as a mapping of the set of points in the plane onto itself. Two mappings a and β of a set S into itself may be combined to yield a third mapping of S into itself, according to the following definition.
Definition: Given two mappings a, β, of a set S into itself, we define a third mapping γ of S into itself by the rule: If y = (x)a and z = (y)β, then z = (x)a. The mapping γ is called the product of a and β, and we write γ = aβ.
Here, since y = (x)a is unique and z = (y)β is unique, z = [(x)a]β = (x)γ is defined for every x of S and is a unique element of S.
Theorem 1.2.1. The mappings of a set S into itself satisfy M0, M1, and M3 if multiplication is interpreted to be the product of mappings.
Proof: It has already been noted that M0 is satisfied. Let us consider M1. Let α, β, γ be three given mappings. Take any element x of S and let y = (x)a, z = (y)β, and w = (z)γ. Then (x)[(αβ)γ] = z([gamma) = w, and (x)[α(βγ)] = y(βγ) = w. Since both mappings, (αβ)γ and α(βγ), give the same image for every x in S, it follows that (αβ)γ = α(βγ).
As for M3, let 1 be the mapping such that (x)1 = x for every x in S. Then 1 is a unit in the sense that for every mapping α, α1 = 1α = α.
In general, neither M2 nor M4 holds for mappings. But M4 holds for an important class of mappings, namely, the one-to-one mappings of S onto itself.
Definition: A mapping a of a set S onto T is said to be one-to-one (which we will frequently write 1–1) if every element of T is the image of exactly one element of S. We indicate such a mapping by the notation: α:x [??] y, where x is an element of S, and y is an element of T. We say that S and T have the same cardinal number of elements.
Theorem 1.2.2. The one-to-one mappings of a set S onto itself satisfy M0, M1, M3, and M4.
Proof: Since Theorem 1.2.1 covers M0, M1, and M3, we need only verify M4. If α:x [??] y is a one-to-one mapping of S onto itself, then by definition, for every y of S there is exactly one x of S such that y = (x)a. This assignment of a unique x to each y determines a one-to-one mapping τ: y [??] x of S onto itself. From the definition of τ we see that (x)(ατ) = x for every x in S and y(τα) = y for every y in S. Hence, ατ = τα = 1, and τ is a mapping satisfying the requirements for a-1 in M4.
We call a one-to-one mapping of a set onto itself a permutation. When the given set is finite, a permutation may be written by putting the elements of the set in a row and their images below them. Thus [MATHEMATICAL EXPRESSION OMITTED] and [MATHEMATICAL EXPRESSION OMITTED] are two permutations of the set S (1, 2, 3). Their product is defined to be the permutation [MATHEMATICAL EXPRESSION OMITTED]. Note that the product rule for permutations given here is obtained by multiplying from left to right. Some authors define the product so that multiplication is from right to left.
1.3. Definitions for Groups and Some Related Systems.
We see that, as single operations, the laws governing addition and multiplication are the same. Of these, all but the commutative law are satisfied by the product rule for the one-to-one mappings of a set onto itself. The laws obeyed by these one-to-one mappings are those which we shall use to define a group.
Definition (First Definition of a Group): A group G is a set of elements G(a, b, c, ...) and a binary operation called "product" such that:
G0. Closure Law. For every ordered pair a, b of elements of G, the product ab = c exists and is a unique element of G.
G1. Associative Law. (ab)c = a(bc).
G2. Existence of Unit. An element 1 exists such that 1a = a1 = a for every a of G.
G3. Existence of Inverse. For every a of G there exists an element a-1 of G such that a-1a = aa-1 = 1.
These laws are redundant. We may omit one-half of G2 and G3, and replace them by:
G2.* An element 1 exists such that 1a = a for every a of G.
G3.* For every a of G there exists an element x of G such that xa = 1.
We can show that these in turn imply G2 and G3. For a given a let
xa = 1 and yx = 1,
by G3.*
Then we have
ax = 1(ax) = (yx)(ax) = y[x(ax)] = y[1x] = yx = 1,
so that G3 is satisfied. Also,
a = 1a = (ax)a = a(xa) = a1,
so that G2 is satisfied.
The uniqueness of the unit 1 and of an inverse a-1 are readily established (see Ex. 13). We could, of course, also replace G2 and G3 by the assumption of the existence of 1 and x such that: a1 = a and ax = 1. But if we assume that they satisfy a1 = a and xa = 1, the situation is slightly different.
There are a number of ways of bracketing an ordered sequence a1a2 ... an to give it a value by calculating a succession of binary products. For n = 3 there are just two ways of bracketing, namely, (a1a2)a3 and a1(a2a3), and the associative law asserts the equality of these two products. An important consequence of the associative law is the generalized associative law.
All ways of bracketing an ordered sequence a1a2, ... an to give it a value by calculating a succession of binary products yield the same value.
It is a simple matter, using induction on n, to prove that the generalized associative law is a consequence of the associative law (see Ex. 1).
Another definition may be given which does not explicitly postulate the existence of the unit.
Definition (Second Definition of a Group): A group G is a set of elements G(a, b, ...) such that
1) For every ordered pair a, b of elements of G, a binary product ab is defined such that ab = c is a unique element of G.
2) For every element a of G a unary operation "inverse," a-1, is defined such that a-1 is a unique element of G.
3) Associative Law. (ab)c = a(bc).
4) Inverse Laws. a-1(ab) = b = (ba)a-1.
It is an easy task to show that any set which satisfies the axioms of the first definition also satisfies those of the second. To show the converse, assume the axioms of the second definition and consider the relation:
a-1a = [(a-1a)b]b-1 = (a-1[a(bb-1)] = bb-1.
When a = b, we see that a-1a = aa-1, and consequently the element a-1a = aa-1 is the same for every a in G. Let us call this element "1," so that G3 is satisfied. Also,
1b = (a-1a)b = a-1(ab) = b,
and
b1 = b(aa-1) = (ba)a-1 = b,
and G2 is satisfied. Therefore the two definitions of a group are equivalent.
There is a third definition of a group as follows:
Definition (Third Definition of a Group): A group G is a set of elements G(a, b, ...) and a binary operation a/b such that:
L0. For every ordered pair a, b of elements of G, a/b is defined such that a/b = c is a unique element of G.
L1. a/a = b/b.
L2. a/(b/b) = a.
L3. (a/a)/(b/c) = c/b.
L4. (a/c)/(b/c) = a/b.
In terms of this operation, let us define a unary operation of inverse b-1 by the rule
b-1 = (b/b)/b.
Here
(b-1)-1 = (b-1/b-1)/b-1 = (b-1/b-1)/[(b/b)/b] = b/(b/b) = b,
using in turn L3 and L2. We now define a binary operation of product by the rule
ab = a/b-1.
Then a/b = a/(b-1)-1 = ab-1. Let us write 1 for the common value of a/a = b/b as given by L1. Then L1 becomes aa-1 = 1, whence also for any a, 1 = a-1(a-1)-1 = a-1a. Thus G3 of the first definition holds. In b-1 = (b/b)/b, put b = 1, whence 1-1 = 11-1, and so 1 = 1/1 = 11-1 = 1-1. L2 now becomes a1-1 = a1 = a. By definition b-1 = 1/b = 1b-1, and with b = a-1, this gives (a-1)-1 = 1(a-1)-1, or a = 1a. Thus G2 of the first definition holds. L3 now becomes 1(bc-1)-1 = cb-1, whence (bc-1)-1 = cb-1. In L4, put a = x, b = 1, c = y-1; whence (xy)(1y)-1 = x1-1 = x or (xy)y-1 = x. Now, for any x, y, z, put a = xy, b = z-1, c = y. Then ac-1 = (xy)y-1 = x, and L4 becomes (ac-1)(bc-1)-1 = ab-1, whence (ac-1)(cb-1) = ab-1. But in terms of x, y, z this becomes x(yz) = (xz)z, the associative law G1. Thus this definition of group implies the first definition. But in terms of the first definition if we put ab-1 = a/b, we easily find that the laws L0, -1, -2, -3, -4 are satisfied, and therefore the definitions are equivalent.
There are systems which satisfy some but not all the axioms for a group. The following are the main types:
Definition: A quasi-group Q is a system of elements Q(a, b, c, ...) in which a binary operation of product ab is defined such that, in ab = c, any two of a, b, c determine the third uniquely as an element of Q.
Definition: A loop is a quasi-group with a unit 1 such that 1a = a1 = a for every element a.
Definition: A semi-group is a system S(a, b, c, ...) of elements with a binary operation of product ab such that (ab)c = a (be).
A group clearly satisfies all these definitions. We may, with Kurosch, further define a group as a set which is both a semi-group and a quasi-group. As a semi-group G0 and G1 are satisfied. Let t be the unique element such that tb = b for some particular b, and let y be determined by b and a so that by = a. Then (tb)y = by and t(by) = by, or ta = a for any a, and G2* is satisfied. In a quasigroup G3* is also satisfied. But we have already shown that these properties define a group.
We call a system with a binary product and unary inverse satisfying
a-19ab) = b = (ba)a-1
a quasi-group with the inverse property, this law being the inverse property. We must show that the product defines a quasi-group. If ab = c, we find b = a-1(ab) = a-1c, and a = (ab)b-1 = cb-1. Thus a and b determine c uniquely; and also given c and a, there is at most one b, and given c and b, there is at most one a. Write a(a-1c) = w. Then a-1[a(a-1c)] = a-1w, whence a-1c = a-1w. Then (a-1)-1(a-1c) = (a-1)-1(a-1c) whence c = w. Hence a(a-1c) = c, and similarly, (cb-1)b = c, and the system is a quasi-group. We note that an inverse quasi-group need not be a loop. With three elements a, b, c and relations a2 = a,ab = ba = c, b2 = b, bc = cb = a, c2 = c, ca = ac = b, we find that each element is its own inverse, and we have a quasi-group with inverse property but no unit.
(Continues…)
Excerpted from "The Theory of Groups"
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