Properties of a group
How can we say that the pair (G, o) is a group. Explain the properties which proof it.
Expert
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 thatg o g-1 = g-1 o g = e:There is another property implicit in this denition 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).
Who firstly use the finite-difference method?
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
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
1. Caterer determines that 87% of people who sampled the food thought it was delicious. A random sample of 144 out of population of 5000 taken. 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?<
Let G be a group. (i) G satises 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)
Determine into which of the following 3 kinds (A), (B) and (C) the matrices (a) to (e) beneath can be categorized: Type (A): The matrix is in both reduced row-echelon form and row-echelon form. Type (B): The matrix
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
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?
The ABC Company, a merchandising firm, has budgeted its action for December according to the following information: • Sales at $560,000, all for cash. • The invoice cost for goods purc
In differentiated-goods duopoly business, with inverse demand curves: P1 = 10 – 5Q1 – 2Q2P2 = 10 – 5Q2 – 2Q1 and per unit costs for each and every firm equal to 1.<
18,76,764
1939591 Asked
3,689
Active Tutors
1433838
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!