--%>
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 : 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

    		•
  • Q : Problem on reduced row-echelon The

    The augmented matrix from a system of linear equations has the following reduced row-echelon form. 280_row echelon method.jpg

    		•
  • Q : Properties for polynomial Specify the

    Specify the important properties for the polynomial.

    		•
  • 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 : Use MS Excel to do the computations

    Select a dataset of your interest (preferably related to your company/job), containing one variable and atleast 100 data points. [Example: Annual profit figures of 100 companies for the last financial year]. Once you select the data, you should compute 4-5 summary sta

    		•
  • Q : Area Functions & Theorem Area Functions

    Area Functions 1. (a) Draw the line y = 2t + 1 and use geometry to find the area under this line, above the t - axis, and between the vertical lines t = 1 and t = 3. (b) If x > 1, let A(x) be the area of the region that lies under the line y = 2t + 1 between t

    		•
  • Q : Global And Regional Economic Development

    The Pharmatec Group, a supplier of pharmaceutical equipment, systems and services, has its head office in London and primary production facilities in the US. The company also has a successful subsidiary in South Africa, which was established in 1990. Pharmatec South A

    		•
  • 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 : Profit-loss based problems A leather

    A leather wholesaler supplies leather to shoe companies. The manufacturing quantity requirements of leather differ depending upon the amount of leather ordered by the shoe companies to him. Due to the volatility in orders, he is unable to precisely predict what will b

    		•
  • 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

    		•