Bsc Mathematics Abstract Algebra by sk mapa Group Theory exercise 8 solution pdf

Abstract Algebra exercise 8 solution pdf.

Groups.

A non-empty set G is said to form a group with respect to binary composition o , if

(i) G in closed under the composition,

(ii) o is associative,

(iii) there exists an element e in G such that aoe=eoa=e  for all a in G.

 (iv) for each elementa in G, there exists an element a' in G such that a'oa=aoa'=e

The group in denoted by the symbol (G.,o).

The element e is said to be an identity element in the group. We shall prove that there is only one inverse element in the group and therefore e will be said to be the identity clement.

The element a' is said to be an inverse of a .We shall prove that each element a has only one inverse and therefore  a' will be said to be the inverse of a .

Definition, 

A group (G,o) is said to be a commutative group or an the name of Norwegian mathematician N. Abel) if composition is commutative.

Necessary Theorem:

theorem 1.

 A group (G,o) contains only one identity element.

Theorem 2.

In a group (G) each element has only one immerse.

Theorem 3. 

Cancellation laws. 

In a group (G,o), for all a,b,c€G

(i) aob = aoc implies b = c (left cancellation law);

(ii) boa = coa implies b = c (right cancelation law)

Theorem 4. 

In a group (G,o), for all a, b in G, each of the equation aox = b and yoa = b has a unique solution in G.

Theorem 5.

 In a group (G,.o),(aob) -1 =b^-1oa^-1 for all a, b€ G.

Theorem 6.

 Let (G,o) be a semigroup and for any two elements a, b in G, each of the equation aox=b and yoa= b has a solution in G. Then (G,o) is a group.

Theorem 7. 

Let G be a non-empty set satisfying the conditions -

(1) G is closed under a binary composition o.

 (2) o is associative,

(3) there exists an element e in G such that eoa=a for all a in G .

(4) for each element a in G, there exists an element a' in G such that

aoa'=e

Then (G ) is a group.

Note 1. 

In an analogous manner, a group (G,) can be defined as a non-empty set G together with a binary composition o satisfying the conditions

(1) G is closed under o,

(2) o is associative, 

(3) there exists an element in G such that aoe = a for all a in G.

(6) for each element in G, there exists an element a' in G such that aoa'= e.

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