7.3. Subroups. Homomorphisms

Let be a subset of a group G. Then is called a subgroup of G if is itself a group under the operation of G.

A subset N of a group G is a subgroup of G if :

1. The identity element e e N.

2. N is closed under the operation of G, i.e. if a,^ e N then a^ e N.

3. N is closed under inverse, that is, a e N, then ae N .

Every group G has the subgroups {e } and G itself. Any other subgroup of G is called a nontrivial subgroup.

T^eorew (LagraMge). Let N be a subgroup of a finite group G. Then the order of N divides the order of G.

Exawp/e 7.3.7. Consider the group G of 2 x 2 matrices with rational entries and nonzero detearminants. Let N be the subset of G consisting of matrices whose

a 01

upper-right entry is zero, that is, matrices of the form . Then N is a subgroup

L c J

of G since N is closed under multiplication and inverses and 7 e N .

De/'M'^'oM. A mapping / from a group G into a group G' is called a homomorphis if, for every a,^ e G, /(a^)= /(a)/(^).

In addition, if / is one-to-one and onto, then/ is called an isomorphism; and G and G' are said to be isomorphic, written G = G'.

If /: G ^ G' is a homomorphism, then the kernel of written ^er / is the set of elements whose image is the identity e' of G'; that is,

The number 1 is the identity element and — is the multiplicative inverse of the

^er/ = {a e G/ (a )= e^}. Recall that the image ofwritten / (G) or Im /, consists of the images of the elements under that is, Im / = e Gl there exists a e G for which / (a) =

E.awT'/e 7.3.2. a) Let G be the group of real numbers under addition, and let G' be the group of positive real numbers under multiplication. The mapping /: G ^ G'

defined by /(a) = 2^a is a homomorphism because /(a + A)= 2^a -2^A = /(a)/(A). In fact, / is also one-to-one and onto; hence G and G' are isomorphic. c) Let a be any element in a group G. The function /: Z ^ G G' defined by

/(^)= a" is a homomorphism since /(w + ^)= a^w +" = a^w - a^w = /(w)- /(^).

7.4. Rings. Fields

Let ^ be a non-empty set with two binary operations: an operation of addition and an operation of multiplication. Then ^ is called a ring if the following axioms are satisfied:

1. For any a, A, c e we have (a + A) + c = a + (A + c).

2. There exists an element 0 e ^, called the zero element, such that for every a e ^, a + 0 = 0 + a = a.

3. For each a e ^ there exists an element - a e ^, called the negative of a, such that a + (- a) = (- a) + a = 0 .

4. For any a, A e we have a + A = A + a.

5. For any a, A, c e we have (a - A) - c = a - (A - c).

6. For any a, A, c e we have (i) a - (A + c) = aA + ac, and (ii) (A + c)a = Aa + ca.

Observe that the axioms 1) through 4) may be summarized by saying that ^ is an abelian group under addition.

Subtraction is defined in ^ by a - A = a + (- A).

A subset ^ of ^ is a subring of ^ if ^ itself is a ring under the operations in We note that ^ is a subring of ^ if : (i) 0 e ^, and (ii) for any a, A e ^,we have a - A e ^ and a - A e ^.

De/''"''^''o". ^ is called a commutative ring if aA = Aa for every a, A e ^.

De/''"''^''o". ^ is called a ring with an identity element 1 if the element 1 has the property that a -1 = 1 - a = a for every a e ^. In such a case, an element a e ^ is called a unit if a has a multiplicative inverse, that is, an element a in R such that

a - a = a - a = 1.

De/''"''^''o". R is called a ring with zero divisors if there exist nonzero elements a, A e ^ such that aA = 0. In such a case, a and A are called zero divisors.

De/''"''^''o". A commutative ring ^ is an integral domain if ^ has no zero divisors, that is, aA = 0 implies a = 0 or A = 0.

De/'M'Y'oM. A commutative ring ^ with an identity element 1 (not equal to 0) is a field if every nonzero a E ^ is a unit, that is, has a multiplicative inverse.

A field is necessarily an integral domain, for if a^ = 0 and a # 0, then ^ = 1. We remark that a field may also be viewed as a commutative ring in which the nonzero elements form a group under multiplication.

Exaw^/e 7.4.7.

1. The set Z integers with the usual operations of addition and multiplication is the classical example of an integral domain (with an identity element). The units in Z are only 1 and - 1, that is, no other element in Z has a multiplicative inverse.

2. usual operations of addition and mul c) Let ^ be any ring. Then the set ^

   The rational numbers g and real numbers ^ each forms a field with respect to the

iplication.

x] of all polynomials over ^ is a ring with respect to the usual operations of addition and multiplication of polynomials. Moreover, if ^ is an integral domain then ^[x] is also an integral domain.

De/'M'Y'oM. A subset 7 of a ring ^ is called an ideal in ^ if the following three properties hold:

1. 0 E 7 .

2. For any a, ^ E 7 we have a - ^ E 7.

3. For any r E ^ and a E 7, we have ra, ar E 7 .

Now let ^ be a commutative ring with an identity element. For any a E ^, the following set is an ideal:

(a )={rar E ^}= a^.

Exaw^/e 7.4.2. Let ^ be any ring. Then { 0 } and ^ are ideals. In particular, if ^ is a field, then { 0 } and ^ are the only ideals.