vikas92 said:Homework Statement
Let a and b be elements of a group,with a^2=e , b^6=e and a.b=b^4.a find its order and express its inverse in form of a^m.b^n
Homework Equations
The Attempt at a Solution
(ab)^2=(ab)(ab)=(ab)(b^4.a)=a(b^5)a
(ab)^3=a(b^5)a(ab)=a(b^5)(a^2)b=a(b^6)=ae=a
it implies(ab)^6=e
thus order(ab) divides 6.It is not 3 hence it is 6.
Help me in solving second part.I can express inverse in form of b^m.a^n but not in the asked form.
algebrat said:you want (ab)^(-1)?
try (b^4a)^(-1)
Group theory is a branch of mathematics that studies the algebraic structures known as groups. A group is a set of elements with a binary operation that satisfies certain properties, such as closure, associativity, identity, and invertibility.
The order of an element in a group is the smallest positive integer n such that the element raised to the power of n equals the identity element. This can be found by repeatedly applying the group operation to the element until the result is the identity element.
The inverse of an element in a group is another element that, when combined with the original element using the group operation, results in the identity element. In other words, it "undoes" the original element. Every element in a group has an inverse, and it is unique.
The inverse of an element in a group can be found by using the group's operation to "cancel out" the original element. For example, if the group operation is addition, the inverse of an element x would be -x. If the group operation is multiplication, the inverse of an element x would be 1/x.
Group theory has many applications in various fields, including chemistry, physics, computer science, and cryptography. For example, in chemistry, group theory is used to study the symmetry of molecules. In physics, it is used to describe the behavior of particles in quantum mechanics. In computer science, group theory is used in cryptography to create secure communication protocols.