--%>

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 : How do it? integral e^(-t)*e^(tz) t

    integral e^(-t)*e^(tz) t between 0 and infinity for Re(z)<1

  • Q : Maths assignment complete assignment

    complete assignment with clear solution and explanation

  • 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 : 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 : Numerical Analysis Hi, I was wondering

    Hi, I was wondering if there is anyone who can perform numerical analysis and write a code when required. Thanks

  • Q : Test Please read the assignment

    Please read the assignment carefully and confirm only if you are 100% sure. Please go through below mentioned guidelines and penalties: • Your solution must be accurate and complete. • Please do not change Subject Title of the Email. • Penalty clause will be applied in case of delayed or plag

  • Q : What is Big-O hierarchy The big-O

    The big-O hierarchy: A few basic facts about the big-O behaviour of some familiar functions are very important. Let p(n) be a polynomial in n (of any degree). Then logbn is O(p(n)) and p(n) is O(an<

  • Q : Mean and standard deviation of the data

    Below is the amount of rainfall (in cm) every month for the last 3 years in a particular location: 130 172 142 150 144 117 165 182 104 120 190 99 170 205 110 80 196 127 120 175

  • Q : Problem on inventory merchandise AB

    AB Department Store expects to generate the following sales figures for the next three months:                            

  • Q : Calculus I need it within 4 hours. Due

    I need it within 4 hours. Due time March 15, 2014. 3PM Pacific Time. (Los Angeles, CA)