--%>
Assignment Help - homework help

Examples of groups

Examples of groups: We now start to survey a wide range of examples of groups (labelled by (A), (B), (C), . . . ). Most of these come from number theory. In all cases, the group axioms should be checked. This is easy for almost all of the examples, and will be left as an exercise except in the occasional more difficult or subtle case.

(A) Our first examples are groups of numbers under addition. To begin, each of the sets Z (the integers), Q (the rational numbers), R (the real numbers) and C (the complex numbers) forms a group under the binary operation + of addition (exercise). Clearly, the groups are all abelian.

(B) For any fixed n ≡ Z, the set nZ = {na : a ≡ Z} is a subgroup of Z (exercise). A few speci fic cases are:

0Z = {0};
1Z = ( -1)Z = Z;
2Z = ( -2)Z = {2a : a ≡ Z}
= the set of even integers:

   Related Questions in Mathematics

  • Q : Properties for polynomial Specify the

    Specify the important properties for the polynomial.

    		•
  • Q : Problem on mass balance law Using the

    Using the mass balance law approach, write down a set of word equations to model the transport of lead concentration. A) Draw a compartmental model to represent  the diffusion of lead through the lungs and the bloodstream.

    		•
  • Q : Explain Black–Scholes model Explain

    Explain Black–Scholes model.

    		•
  • Q : Nonlinear integer programming problem

    Explain Nonlinear integer programming problem with an example ?

    		•
  • Q : Budgeted cash disbursements The ABC

    The ABC Company, a merchandising firm, has budgeted its action for December according to the following information: • Sales at $560,000, all for cash. • The invoice cost for goods purc

    		•
  • Q : Containee problem For queries Q 1 and Q

    For queries Q1 and Q2, we say Q1 is containedin Q2, denoted Q1 C Q2, iff Q1(D) C Q2

    		•
  • Q : Explain Factorisation by Fermats method

    Factorisation by Fermat's method: This method, dating from 1643, depends on a simple and standard algebraic identity. Fermat's observation is that if we wish to nd two factors of n, it is enough if we can express n as the di fference of two squares.

    		•
  • Q : Problem on Nash equilibrium In a

    In a project, employee and boss are working altogether. The employee can be sincere or insincere, and the Boss can either reward or penalize. The employee gets no benefit for being sincere but gets utility for being insincere (30), for getting rewarded (10) and for be

    		•
  • Q : Explain Factorisation by trial division

    Factorisation by trial division: The essential idea of factorisation by trial division is straightforward. Let n be a positive integer. We know that n is either prime or has a prime divisor less than or equal to √n. Therefore, if we divide n in

    		•
  • Q : Ordinary Differential Equation or ODE

    What is an Ordinary Differential Equation (ODE)?

    		•