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.





(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 : The mean of the sampling distribution

    1. Caterer determines that 87% of people who sampled the food thought it was delicious. A random sample of 144 out of population of 5000 taken. The 144 are asked to sample the food. If P-hat is the proportion saying that the food is delicious, what is the mean of the sampling distribution p-hat?<

  • Q : Define terms Terms : Terms are defined

    Terms: Terms are defined inductively by the following clauses.              
    (i) Every individual variable and every individual constant is a term. (Such a term is called atom

  • 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 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 : Properties of a group How can we say

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

  • Q : Problem on Datalog for defining

    The focus is on  the use of Datalog for defining properties  and queries on graphs.

    (a) Assume that P is some property of graphs  definable in the Datalog. Show that P is preserved beneath extensions  and homomo

  • Q : Probability assignments 1. Smith keeps

    1. Smith keeps track of poor work. Often on afternoon it is 5%. If he checks 300 of 7500 instruments what is probability he will find less than 20substandard?

    2. Realtors estimate that 23% of homes purchased in 2004 were considered investment properties. If a sample of 800 homes sold in 2

  • Q : State Prime number theorem Prime number

    Prime number theorem: A big deal is known about the distribution of prime numbers and of the prime factors of a typical number. Most of the mathematics, although, is deep: while the results are often not too hard to state, the proofs are often diffic

  • Q : Problem on mixed-strategy equilibrium

    Assume three Offices (A, B, & C) in downtown,  simultaneously decide whether to situate in a new Building. The payoff matrix is illustrated below. What is (are) the pure stratgy Nash equilibrium (or equilibria) and mixed-strtegy equilibrium of the game?

  • Q : Simulation with Arena An office of

    An office of state license bureau has two types of arrivals. Individuals interested in purchasing new plates are characterized to have inter-arrival times distributed as EXPO(6.8) and service times as TRIA(808, 13.7, 15.2); all times are in minutes. Individuals who want to renew or apply for a new d

2015 ©TutorsGlobe All rights reserved. TutorsGlobe Rated 4.8/5 based on 34139 reviews.