--%>
Assignment Help - homework help

Properties of a group

How can we say that the pair (G, o) is a group. Explain the properties which proof it.

E

Expert

Verified

Let G be a set and suppose that o is a binary operation on G. We say that the pair (G; o) is a group if it has the following properties.

(i) The operation o is associative; that is, (g o h) o k = g o (h o k) for all g; h; k ≡ G.

(ii) There exists an identity for o ; that is, there exists e ≡ G such that g o e = e o g = g for all g ≡ G.

(iii) There exist inverses for o ; that is, for each g ≡ G, there exists g-1≡ G such that
g o g-1 = g-1 o g = e:

There is another property implicit in this de nition which it is useful to give a name to. Instead of saying that o is a binary operation on G, we can say that the law of closure holds for o, meaning that when o acts on two elements of G the result is also in G.

Most of the groups (G; o) we study will also have the following property.

(iv) The operation o is commutative; that is, g o h = h o g for all g; h ≡ G.

A group with this property is called commutative or, more usually, abelian, after the Norwegian mathematician Niels Henrik Abel (1802{1829).

   Related Questions in Mathematics

  • Q : Mathematical Method for Engineers The

     The function is clearly undefined at , but despite all of this the function does have a limit as approaches 0. a) Use MATLAB and ezplot to sketch for , and use the zoom on facility to guess the . You need to include you M-file, outp

    		•
  • Q : Who derived the Black–Scholes Equation

    Who derived the Black–Scholes Equation?

    		•
  • 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 : Formal logic It's a problem set, they

    It's a problem set, they are attached. it's related to Sider's book which is "Logic to philosophy" I attached the book too. I need it on feb22 but feb23 still work

    		•
  • Q : Probability and Stochastic assignment

    Introduction to Probability and Stochastic Assignment 1: 1. Consider an experiment in which one of three boxes containing microchips is chosen at random and a microchip is randomly selected from the box.

    		•
  • Q : Statistics math Detailed explanation of

    Detailed explanation of requirements for Part C-1 The assignment states the following requirement for Part 1, which is due at the end of Week 4: “Choose a topic from your field of study. Keep in mind you will need to collect at least [sic] 3- points of data for this project. Construct the sheet y

    		•
  • Q : Formulating linear program of a

    A software company has a new product specifically designed for the lumber industry. The VP of marketing has been given a budget of $1,35,00to market the product over the quarter. She has decided that $35,000 of the budget will be spent promoting the product at the nat

    		•
  • 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 : Law of iterated expectations for

     Prove the law of iterated expectations for continuous random variables. 2. Prove that the bounds in Chebyshev's theorem cannot be improved upon. I.e., provide a distribution that satisfies the bounds exactly for k ≥1, show that it satisfies the bounds exactly, and draw its PDF. T

    		•
  • 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.

    		•