Machì - Groups: an Introduction to Ideas and Methods of the Theory of Groups

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Antonio Mach UNITEXT Groups An Introduction to Ideas and Methods of the Theory of Groups 10.1007/978-88-470-2421-2_1 Springer-Verlag Italia 2012

1. Introductory Notions

Antonio Mach 1

(1)

Department of Mathematics, University La Sapienza, Rome, Italy

Abstract

Definition 1.1 . A group G is a non empty set in which it is defined a binary operation, i.e. a function:

such that, if ab denotes the image of the pair ( a, b ),

i)

the operation is associative: ( ab ) c = a ( bc ), for all triples of elements a, b, c G ;

ii)

there exists an element e G such that ea = a = ae , for all a G . This element is unique: if e is also such that ea = a = ae , for all a G , ea = a implies, with a = e , that ee = e , and a = ae implies, with a = e , that ee = e . Thus e = e ;

iii)

for all a G , there exists b G such that ab = e = ba .

1.1 Definitions and First Theorems

Definition 1.1.

A group G is a non empty set in which it is defined a binary operation, i.e. a function:

such that, if ab denotes the image of the pair ( a , b ),

i)

the operation is associative: ( ab ) c = a ( bc ), for all triples of elements a , b , c G ;

ii)

there exists an element e e G such that ea = a = ae , for all a e G . This element is unique: if e is also such that e a = a = ae , for all a e G , ea = a implies, with a = e , that ee = e , and a = ae implies, with a = e, that ee = e. Thus e = e;

iii)

for all a G , there exists b G such that ab = e = ba .

We then say that the above operation endows the set G with a group structure .

The set G underlies the group defined in it. The element ab is the product of the elements a and b (in this order); one also writes a b , a * b, a b , a + b and the like, and G (), G (+) if the operation is to be emphasized.

The element e is the identity element of the operation (it leaves an element unchanged when combined with it). Other names are neutral element , unity , zero (the latter when the operation is denoted additively); we shall mainly use the notation 1, I or 0 (zero). If the group G reduces to a single element, necessarily the identity, then G is the trivial group or the identity group , denoted G = {1} or G = {0}.

From i ), ii ) and iii ) follows the cancellation law :

Indeed, if ab = ac , let x G be such xa = e; from x ( ab ) = x ( ac ) it follows, by i ), ( xa ) b = ( xa ) c , so that eb = ec and b = c . The other implication is similar. In particular, if ax = e = ay , then x = y , that is, given a , the element x of iii ) is unique. This element is called the inverse of a , and is denoted a 1 (in additive notation it is the opposite of a , denoted a ). From aa 1 = e it follows that a is an inverse for a 1, and therefore the unique one; hence ( a 1)1 = a . The cancellation law implies that multiplication by an element of the group is a bijection , meaning that if S is a subset of G , and x an element of G , the correspondence S Sx between S and the set of products Sx = { sx , s S } given by s sx is a bijection. It is onto ( sx comes from s ), and it is injective because if sx = s x , then by the cancellation law s = s . In particular, with S = G we obtain, for each x G , a bijection of G with itself, that is, a permutation of the set G (see Theorem 1.8).

When a binary operation is defined in a set, and this operation is associative, then one can speak of the product of any number of elements in a fixed order, meaning that all the possible ways of performing the product of pairs of elements yield the same result. For instance, a product of five elements a , b , c , d , e , in this order, can be performed as follows: first ab = u and cd = , then uv , and finally the product of this element with e : ( uv ) e . On the other hand, one can first perform the products bc = x and de = y, then the product xy , and finally the product of a with the latter element, a ( xy ). By the associative law, a ( xy ) = ( ax ) y , i.e. ( a ( bc )) y = (( ab ) c ) y = ( uc ) y , and ( uc ) y = u ( cy ) = u ( c ( de )) = u (( cd ) e ) = u ( ve ) = ( uv ) e , as above.

We now prove this in general.

Theorem 1.1 (Generalized associative law).

All the possible products of the elements a 1, a 2,, a n of a group G taken in this order are equal .

Proof . Induction on n . For n = 1, 2 there is nothing to prove. Let n 3; for n = 3 the result is true because the operation is associative. Let n > 3, and assume the result true for a product of less than n factors. Any product of the n elements is obtained by multiplying two elements at a time, so that it will eventually reduce to a product of two factors: ( a 1 a r ) ( a r +1 a n ). If ( a 1 a s ) ( a s +1 a n ) is another such reduction, we may assume without loss that r s . By induction,

is a well defined product, as well as

It follows:

and the associativity of the operation gives the result.

In Particular, if a i = a for all i , we denote the product by a n , and for m , n we have :

setting a 0 = e . Writing a m = ( a 1) m (the product of m times a 1), the first equality also holds for negative m and n ; if only one is negative, the same equality is obtained by deleting the elements of type aa 1 (or a 1 a ) that appear. Moreover,

and

Thus, the above equalities hold in all cases, with positive, negative or zero exponents; in additive notation they become:

Moreover,

and therefore

More generally,

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