--%>

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 : Problem on sales and budget XYZ Farm

    XYZ Farm Supply data regarding the store's operations follow: • Sales are budgeted at $480,000 for November, $430,000 for December, and $340,000 for January. • Collections are expected

  • Q : Examples of groups 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, an

  • Q : What is the definition of a group Group

    Group: Let G be a set. When we say that o is a binary operation on G, we mean that o is a function from GxG into G. Informally, o takes pairs of elements of G as input and produces single elements of G as output. Examples are the operations + and x of

  • Q : Numerical solution of PDE i want you to

    i want you to solve this assignment. this consist of two parts theoretical and coding. the code has to be created by you. no modified or copying code. you have to mention the exact solution and the proportion error. also you have to explain the sketch that you get from the code. these information

  • Q : Theorem-Group is unique and has unique

    Let (G; o) be a group. Then the identity of the group is unique and each element of the group has a unique inverse.In this proof, we will argue completely formally, including all the parentheses and all the occurrences of the group operation o. As we proce

  • Q : Problem on augmented matrix Consider

    Consider the following system of linear equations.  (a) Write out t

  • Q : What is limit x tends to 0 log(1+x)/x

    What is limit x tends to 0  log(1+x)/x to the base a?

  • Q : Who firstly use the finite-difference

    Who firstly use the finite-difference method?

  • Q : Explain the work and model proposed by

    Explain the work and model proposed by Richardson.

  • 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.