What is the order of generator?

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.

How do you find the generator of a group?

If the order of a group is 8 then the total number of generators of group G is equal to positive integers less than 8 and co-prime to 8. The numbers 1, 3, 5, 7 are less than 8 and co-prime to 8, therefore if a is the generator of G, then a3,a5,a7 are also generators of G.

What are the two groups of order 4?

There exist exactly 2 groups of order 4, up to isomorphism: C4, the cyclic group of order 4. K4, the Klein 4-group.

What is a generator in group theory?

A set of generators. is a set of group elements such that possibly repeated application of the generators on themselves and each other is capable of producing all the elements in the group. Cyclic groups can be generated as powers of a single generator.

Is a group of order 4 Abelian?

The Klein four-group, with four elements, is the smallest group that is not a cyclic group. There is only one other group of order four, up to isomorphism, the cyclic group of order 4. Both are abelian groups.

Is the Klein 4 group Abelian?

Klein Four Group It is smallest non-cyclic group, and it is Abelian. Klein four group is the symmetry group of a rhombus (or of a rectangle, or of a planar ellipse), with the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180 degree rotation.

What is an example of a generator?

The definition of a generator is someone or something that produces something. An example of a generator is a machine that can produce electrical energy when the power is out in an area. A machine for changing mechanical energy into electrical energy; dynamo.

What is a generator of a group?

The simplest case of a generator is a single element which can literally generate all the elements of the group. One way to describe a group is by means of “generators and relations”. A generator is any operator that is not derivable from the other generators.

What is the meaning of cyclic group of order 4?

Verbal definition. The cyclic group of order 4 is defined as a group with four elements where where the exponent is reduced modulo . In other words, it is the cyclic group whose order is four.

How do you find the Order of the product of generators?

Since the orders of each a i are coprime, the order of their product is equal to the product of their orders, that is a 1… a n has order q 1 k 1… q n k n = p − 1 and thus is a generator. We have ϕ ( q i k i) choices for each a i thus we have exactly ϕ ( q 1 k 1)… ϕ ( q n k n) = ϕ ( p − 1) different generators.

What is the symmetric group of order 24?

The symmetric group is contained in higher symmetric groups, most notably the symmetric group on five elements . These include whose inner automorphism group is (specifically is the quotient of by its scalar matrices). This finite group has order 24 and has ID 12 among the groups of order 24 in GAP’s SmallGroup library.