Define a cyclic group, Mathematics

Assignment Help:

Question 1:

(a) Show that, for all sets A, B and C,

(i) (A ∩ B) c = Ac∩Bc.

(ii) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

(iii) A - (B ∪ C) = (A - B) ∩ (A - C).

(b) Let f: X → Y be a function and Ai ⊂ X for i ∈ I. Show that

(i) f(∩Ai) _∪f(Ai).

(ii) f(∪Ai) = ∪f(Ai).

Question 2:

(a) Prove that the set G = {a + b√3 : a; b ∈ Z} forms an abelian group under ordinary addition.

(b) Let R be a relation in the set of all natural numbers N defined by " xRy if and only if x - y is divisible by 3." Show that R is an equivalence relation. Hence determine the equivalence classes of R.

Question 3:

(a)    A set G consists of ordered pairs (a, b) such that a, b are real numbers and a ≠ 0. An operation * is defined in G as follows

(a, b) * (c, d) = (ac, ad + b) where a, b, c, d are real numbers. Prove that the given set constitutes a non-abelian group for the given operation.

(b) Let (G,*) be a group. Prove the following:

(i) the identity element e is unique,

(ii) (a * b)-1 = b-1 * a-1 for a, b ∈ G.

Question 4:

(a) Prove that a non-empty subset H is a subgroup of (G,*) if and only if

(1) a, b ∈ H ) a* b ∈ H.

(2) a ∈ H ) a-1 ∈ H where a-1 is the inverse of a in G.

(b) Prove that the set Q+ of all positive rational numbers forms an abelian group for the operation * defined by a * b = ab/ 2.

Question 5:

(a) (i) Define a cyclic group.

(ii) Prove that any cyclic group is abelian.

(iii) The set G = {1,-1, I,-i} forms a cyclic group under multiplication of complex numbers.

  • State a generator of G and show how it generates each element of G.
  • Draw the Cayley table for G.

(b) Let H and K be two subgroups of a group G. Show that H ∩ K is a subgroup of G.

 


Related Discussions:- Define a cyclic group

Ratio-categories of situations requiring division , Ratio - situations in ...

Ratio - situations in which we need to compare two quantities in terms of their ratio. (e.g., if Munna weighs 40 Kg. and Munni weighs 50 Kg., find the ratio of their weights.)

Product and quotient rule, Product and Quotient Rule : Firstly let's se...

Product and Quotient Rule : Firstly let's see why we have to be careful with products & quotients.  Assume that we have the two functions f ( x ) = x 3   and g ( x ) = x 6 .

Define multiplication rule in probability, Q. Define Multiplication Rule in...

Q. Define Multiplication Rule in probability? Ans. A family has two girls, Ann and Barb, and three boys, Carl, David and Earl, in it. In how many ways can the mother pick

5th grade, 6 and 3/8 minus 1 and 3/4

6 and 3/8 minus 1 and 3/4

Permuation and combination, how many words can be formed from letters of wo...

how many words can be formed from letters of word daughter such that each word contain 2vowles and 3consonant

Find the height of the building, A building is in the form of a cylinder su...

A building is in the form of a cylinder surrounded by a hemispherical vaulted dome and contains   41(19/21-) cu m of air. If the internal diameter of the building is equal to its t

The bionomial theorem for rational index, use the bionomial theorem to expa...

use the bionomial theorem to expand x+2/(2-X)(WHOLE SQUARE 2)

#title, how do you do hard math!!!

how do you do hard math!!!

Show trigonometric functions on a graph, Q. Show Trigonometric Functions on...

Q. Show Trigonometric Functions on a Graph? Ans. By discussing the trig functions with respect to an angle in a right-angle triangle, we have only considered angles betwee

Line plots, how to you find the difference between different line plots

how to you find the difference between different line plots

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd