## Order of Group in Discrete MathematicsThe term order can be specified into two closely related senses in the group theory: - The group"s order can be described by its cardinality, which means the number of its elements.
- Sometimes, an order can be called a period. So in a group, the order of element 'a" is the positive integer 'm". The relation between a and m is described as follows:
a ^{m} = e
If the order of a group does not have m, a will be known to have the infinite order. In the finite group, all elements will have finite order.
There are 6 elements in the above group. So ord(S ## Order and StructureThe structure of a group can be described using the order of a group and an element. In other words, the group will be complicated if the factorization of the order is complicated. The group will be known as the trivial group if the order of a group is 1. Suppose we have an element x, then x will be called identity only if the ord(x) = 1. The order of group ord(x) will be 2 if every element of x group is equal to its inverse, and consequently, group G will be abelian. Since xy = (yy)xy(xx) = y(yx)(yx)x = yx. If we are trying to converse this statement, it will not be true. The two concepts of order have some relationship, which is described as follows: <a> = {a ^{k} : k ∈ Z}If a generates the subgroup, then ord(a) = ord(<a>). For any integer k, if we have ord(a) divides k, only then we will have the following relation: a ^{k} = eIn general, we can say that the order of G is divided by the order of subgroup of G. More precisely: if the group G has subgroup H, then they will have the following relation: ord(G) / ord(H) = [G : H]
The above relation will generate some immediate consequences in which the order of a group will be divided by the order of every group"s element. For example, in the above symmetric group, we have seen that the ord(S For the finite group, the below partial converse will be true. If d is a prime number and it divides the group"s order G, there will be an element of order d in G. This is known as All power of a will be in finite order if 'a" has finite order. Now we will describe the formula for order of power 'a" if it has finite order: ord(a ^{k} ) = ord(a) / gcd(ord(a), k)Where
The order of inverse of a(a We have no formula that can explain the order of a, b, and the order of product ab. There can be a case in which ab has infinite order, but a and b both have finite order. There can be another case in which ab has finite order, but a and b both have infinite order. If a and b has the relation ab = ba, it will be possible that ord(ab) divides lcm(ord(a), odr(b)). This statement generates consequences, which can prove that if m is used to specify the maximum of all order of group"s element in an abelian group, then every order of element divides m. ## Counting by Order of ElementTo explain counting by order of elements, we will assume that G is used to specify the finite group of order n and d is used to specify the divisor of n. The element"s number in G of order d is a multiplication of φ(d). Where φ is used to specifying the Euler"s totient function, which is used to provide the number of positive integers, but the value of integer will not be larger than d. For example, we have a case S ## In relation to homomorphismsThe order of elements can be reduced with the help of group homomorphisms. If homomorphisms are shown by relation f: G → H and 'a" is an element of G and has a finite order, then ord(a) will be divided by ord(f(a)). The ord(f(a)) will be equal to ord(a) or ord(f(a)) = ord(a) if f is injective. We can use this statement to prove that two concrete given groups have not contained homomorphisms(injective). For example, the relation h: S3 → Z5 has no nontrivial homomorphism because each and every number in Z5 except 0 contains the order 5. In S3, the number 0 can"t divide the order 1, 2, and 3 of elements. There are some further consequences of this through which conjugate elements will contain the same order. ## Class EquationThe class equation can be described as the important result about orders. It is used to relate the finite group"s order G to its center"s order Z(G) and the size of its non-trivial conjugacy classes, which is described as follows: |G| = |Z(G)| + ∑ _{i} d_{i}Where d |S _{3} | = 1+2+3 |