--%>
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 : 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 : 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 : Properties of a group How can we say

    How can we say that the pair (G, o) is a group. Explain the properties which proof it.

    		•
  • Q : Linear programming model of a Cabinet

    A cabinet company produces cabinets used in mobile and motor homes. Cabinets produced for motor homes are smaller and made from less expensive materials than those for mobile homes. The home office in Dayton Ohio has just distributed to its individual manufacturing ce

    		•
  • Q : Theorem-G satis es the right and left

    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)

    		•
  • Q : Who independently developed

    Who independently developed a model for simply pricing risky assets?

    		•
  • Q : Statistics Caterer determines that 37%

    Caterer determines that 37% of people who sampled the food thought it was delicious. A random sample of 144 out of population of 5000. The 144 are asked to sample the food. If P-hat is the proportion saying that the food is delicious, what is the mean of the sampling distribution p-hat?

    		•
  • Q : Formulating linear program of a

    A software company has a new product specifically designed for the lumber industry. The VP of marketing has been given a budget of $1,35,00to market the product over the quarter. She has decided that $35,000 of the budget will be spent promoting the product at the nat

    		•
  • Q : State Prime number theorem Prime number

    Prime number theorem: A big deal is known about the distribution of prime numbers and of the prime factors of a typical number. Most of the mathematics, although, is deep: while the results are often not too hard to state, the proofs are often diffic

    		•
  • Q : Ordinary Differential Equation or ODE

    What is an Ordinary Differential Equation (ODE)?

    		•