--%>

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 : Formulating linear program of an oil

    An oil company blends two input streams of crude oil products alkylate and catalytic cracked to meet demand for weekly contracts for regular (12,000 barrels) mind grade ( 7,500) and premium ( 4,500 barrels) gasoline’s . each week they can purchase up to 15, 000

  • Q : Abstract Boolean Algebra I. Boolean

    I. Boolean Algebra Define an abstract Boolean Algebra, B,  as follows:  The three operations are:  +   ( x + y addition) ( x y multiplic

  • Q : Explain Factorisation by trial division

    Factorisation by trial division: The essential idea of factorisation by trial division is straightforward. Let n be a positive integer. We know that n is either prime or has a prime divisor less than or equal to √n. Therefore, if we divide n in

  • Q : Logic and math The homework is attached

    The homework is attached in the first two files, it's is related to Sider's book, which is "Logic for philosophy" I attached this book too, it's the third file.

  • Q : Explain the work and model proposed by

    Explain the work and model proposed by Richardson.

  • Q : Research Areas in Medical Mathematical

    Some Research Areas in Medical Mathematical Modelling:1. Modeling and numerical simulations of the nanometric aerosols in the lower portion of the bronchial tree. 2. Multiscale mathematical modeling of

  • Q : First-order formulas over the

    Consider the unary relational symbols P and L, and the binary relational symbol On, where P(a) and I(a) encode that a is apoint and a (sraight) line in the 2-dimensional space, respectively, while On(a,b) encodes  that a is a point, b is a line, and o lies on b.

  • Q : Bolzano-Weierstrass property The

    The Bolzano-Weierstrass property does not hold in C[0, ¶] for the infinite set A ={sinnx:n<N} : A is infinite; Show that has no “ limit points”.

  • 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 : 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