Let N, denote a set of n distinct symbols that, for convenience, we represent as{1,2, ..., n}. A permutation of n distinct symbols is a one-to-one mapping from N,to N. Define S, to be the set of all permutations of n distinct symbols. Eachelement of S, is represented by a permutation of the integers 7 in 1, 2, ..., n. It iseasy to demonstrate that S, is a group:

Answered: Image /qna-images/answer/711cf4b8-c241-40c6-ab7f-dbb195222af3.jpg

Answered: Let N, denote a set of n distinct…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.4: Binary Operations

Problem 15E: 15. Let be a binary operation on the non empty set . Prove that if contains an identity element...

See similar textbooks

Similar questions

4. Let be the relation “congruence modulo 5” defined on as follows: is congruent to modulo if and only if is a multiple of , and we write . a. Prove that “congruence modulo ” is an equivalence relation. b. List five members of each of the equivalence classes and .
9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of all even integers is closed with respect to a. addition defined on . b. multiplication defined on .
5. Let be the relation “congruence modulo ” defined on as follows: is congruent to modulo if and only if is a multiple of , we write . a. Prove that “congruence modulo ” is an equivalence relation. b. List five members of each of the equivalence classes and .
19. Let and be nonzero integers. Prove that if and only if divides .
12. (See Exercise 10 and 11.) If each is identified with in prove that . (This means that the order relation defined in Exercise 10 coincides in with the original order relation in . We say that the ordering in is an extension of the ordering in .) 11. (See Exercise 10.) According to Definition 5.29, is defined in by if and only if . Show that if and only if . 10. An ordered field is an ordered integral domain that is also a field. In the quotient field of an ordered integral domain define by . Prove that is a set of positive elements for and hence, that is an ordered field. Definition 5.29 Greater than Let be an ordered integral domain with as the set of positive elements. The relation greater than, denoted by is defined on elements and of by if and only if . The symbol is read “greater than.” Similarly, is read “less than.” We define if and only if. As direct consequences of the definition, we have if and only if and if and only if . The three properties of in definition 5.28 translate at once into the following properties of in . If and then . If and then . For each one and only one of the following statements is true: . The other basic properties of are stated in the next theorem. We prove the first two and leave the proofs of the others as exercises.

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS