Groups
Let S be a set. A mapping $$ S \times S \to S $$ is sometimes called a law of composition (of S into itself). If x, y are elements of S, the image of the pair (x, y) under this mapping is also called their product under the law of composition, and will be
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I
Groups
§1.
MONOIDS Let S be a set. A mapping SxS-+S
is sometimes called a law of composition (of S into itself). If x, yare elements of S, the image of the pair (x , y) under this mapping is also called their product under the law of composition, and will be denoted by xy . (Sometimes, we also write x . y, and in many cases it is also convenient to use an additive notation, and thus to write x + y. In that case, we call this element the sum of x and y. It is customary to use the notation x + y only when the relation x + y = Y + x holds.) Let S be a set with a law of composition. If x, y, z are elements of S, then we may form their product in two ways : (xy)z and x(yz). If (xy)z = x(yz) for all x, y. z in S then we say that the law of composition is associative. An element e of S such that ex = x = xe for all XES is called a unit element. (When the law of composition is written additively, the unit element is denoted by 0, and is called a zero element.) A unit element is unique, for if e' is another unit element, we have e = ee' = e' by assumption. In most cases, the unit element is written simply 1 (instead of e). For most of this chapter, however, we shall write e so as to avoid confusion in proving the most basic properties. A monoid is a set G, with a law of composition which is associative, and having a unit element (so that in particular, G is not empty).
3 S. Lang, Algebra © Springer Science+Business Media LLC 2002
4
I, §1
GROUPS
Let G be a monoid, and x I' . , . , x, elements of G (where n is an integer > I). We define their product inductively: n
TIx
v
= XI " 'X n = (xI · ··Xn-l)xn·
v= I
We then have thefollowing rule: m
m +n
n
TI xI" TI
1'=1
Xm+v
v=1
TI
=
XV'
v=1
which essentially asserts that we can insert parentheses in any manner in our product without changing Its value. The proof is easy by induction, and we shall leave it as an exercise. One also writes m+ n
TI x,
n
TI
instead of
Xm + v
v= I
m+l
and we define o
TI x, = e. v= 1
As a matter of convention, we agree also that the empty product is equal to the unit element. It would be possible to define more general laws of composition, i.e. maps SI x S2 -+ S3 using arbitrary sets. One can then express associativity and commutativity in any setting for which they make sense. For instance, for commutativity we need a law of composition f:S x S
-+
T
where the two sets of departure are the same. Commutativity then means f(x, y) = f(y, x), or xy = yx if we omit the mapping j from the notation. For associativity, we leave it to the reader to formulate the most general combination of sets under which it will work. We shall meet special cases later, for instance arising from maps S x S
-+
Sand
S x T
-+
T.
Then a product (xy)z makes sense with XES, YES, and z E T. The product x(yz) also makes sense for such elements x, y, z and thus it makes sense to say that our law of composition is associative, namely to say that for all x, y, z as above we have (xy)z = x(yz). If the law of composition of G is commutative, we als
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