--%>
Assignment Help - homework help

Theorem-G satis es the right and left cancellation laws

Let G be a group.

(i) G satis es the right and left cancellation laws; that is, if a; b; x ≡ G, then ax = bx and xa = xb each imply that a = b.

(ii) If g ≡ G, then (g-1)-1 = g.

E

Expert

Verified

Proof:

(i) From ax = bx, we have axx-1 = bxx-1, then ae = be, then a = b. Similarly for the other case.

(ii) Temporarily denote the inverse of g-1 by h (instead of (g-1)-1). Then the defining property of h, from the axiom for inverses applied to g-1, is that

g-1h = hg-1 = e:

But g itself satis es these equations in place of h, because the axiom for inverses applied to g says that

gg-1 = g-1g = e:

Hence, since inverses are unique, h = (g-1)-1 = g, as required.

   Related Questions in Mathematics

  • Q : What is Non-Logical Vocabulary

    Non-Logical Vocabulary: 1. Predicates, called also relation symbols, each with its associated arity. For our needs, we may assume that the number of predicates is finite. But this is not essential. We can have an infinite list of predicates, P

    		•
  • Q : Who derived the Black–Scholes Equation

    Who derived the Black–Scholes Equation?

    		•
  • Q : Relationships Between Data Introduction

    Relationships Between Data - Introduction to Linear Regression Simple Regression Notes If you need guidance in terms of using Excel to run regressions, check pages 1 - 10 of the Excel - Linear Regression Tutorial posted to th

    		•
  • Q : Problem on inverse demand curves In

    In differentiated-goods duopoly business, with inverse demand curves: P1 = 10 – 5Q1 – 2Q2P2 = 10 – 5Q2 – 2Q1 and per unit costs for each and every firm equal to 1.<

    		•
  • Q : State Fermat algorithm The basic Fermat

    The basic Fermat algorithm is as follows: Assume that n is an odd positive integer. Set c = [√n] (`ceiling of √n '). Then we consider in turn the numbers c2 - n; (c+1)2 - n; (c+2)2 - n..... until a perfect square is found. If th

    		•
  • Q : Problem on budgeted cash collections

    XYZ Company collects 20% of a month's sales in the month of sale, 70% in the month following sale, and 5% in the second month following sale. The remainder is not collectible. Budgeted sales for the subsequent four months are:     

    		•
  • Q : Problem on Prime theory Suppose that p

    Suppose that p and q are different primes and n = pq. (i) Express p + q in terms of Ø(n) and n. (ii) Express p - q in terms of p + q and n. (iii) Expl

    		•
  • Q : Explain the work and model proposed by

    Explain the work and model proposed by Richardson.

    		•
  • Q : Numerical solution of PDE this

    this assignment contains two parts theoretical and coding the code has to be a new. old code and modified code will appear in the university website .

    		•
  • Q : Who independently developed

    Who independently developed a model for simply pricing risky assets?

    		•