What order are the 4 elements?

What order are the 4 elements?

The Four Elements. Greek philosophy supposed the Universe to comprise four elements: Fire, Air, Earth, & Water. The Four Elements are usually arranged as four corners, but can also be arranged in ascending order, from bottom to top, the Earth rising out of Water, Air over the Earth, and the Sun (Fire) over all.

What is the order of a group element?

If the group is seen multiplicatively, the order of an element a of a group, sometimes also called the period length or period of a, is the smallest positive integer m such that am = e, where e denotes the identity element of the group, and am denotes the product of m copies of a.

What is the largest order of an element in SN?

In mathematics, Landau’s function g(n), named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric group Sn.

What is the order of a multiplicative group?

The concept of multiplicative order is a special case of the order of group elements. The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n.

What is the order of a cyclic group?

Definition and notation The order of g is the number of elements in ⟨g⟩; that is, the order of an element is equal to the order of its cyclic subgroup. A cyclic group is a group which is equal to one of its cyclic subgroups: G = ⟨g⟩ for some element g, called a generator.

Is Z +) a cyclic group?

The integers Z under ordinary addition are a cyclic group, being generated by 1 or −1. Every subgroup of (Z, +) is cyclic. More, precisely, if I is a non-zero subgroup of (Z, +), then I is generated by the smallest integer n in I, i.e, I = nZ = {kn|k ∈ Z}.

What is the order of Z11?

〈(4,3)〉 = {(0,0),(4,3),(8,6),(0,9),(4,12),(8,15)}. So the order of (Z12 × Z18)/〈(4,3)〉 is (12 × 18)/6 = 36. Solution: It is easy to see that 〈(1,1)〉 = Z11 × Z15. So the order of (Z11 × Z15)/〈(1,1)〉 is 1.

Is 2 a generator of Z11?

2 is a generator and thus Z11 is cyclic.

What’s a subgroup?

A subgroup is a subset of group elements of a group. that satisfies the four group requirements. It must therefore contain the identity element. “

What makes a group Abelian?

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

What is the smallest Abelian group?

The smallest noncyclic group is the four element Klein four-group https://en.wikipedia.org/wiki/Klein_four-group . All finite abelian groups are products of cyclic groups. If the factors have orders that are not relatively prime the result won’t be cyclic.