--%>
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 : 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 : Where would we be without stochastic

    Where would we be without stochastic or Ito^ calculus?

    		•
  • 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 : Define Big-O notation Big-O notation :

    Big-O notation: If f(n) and g(n) are functions of a natural number n, we write f(n) is O(g(n)) and we say f is big-O of g if there is a constant C (independent of n) such that f

    		•
  • Q : Uniform scaling what is uniform scaling

    what is uniform scaling in computer graphic

    		•
  • Q : Graph Theory is the n-Dimensional Qn

    is the n-Dimensional Qn Hamiltonian? Prove tour answer

    		•
  • Q : Maths A cricketer cn throw a ball to a

    A cricketer cn throw a ball to a max horizontl distnce of 100m. If he throws d same ball vertically upwards then the max height upto which he can throw is????

    		•
  • 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 : 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

    		•