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math-chick_41
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in a non abelian group does (ab)^2 = (b^2)(a^2)?
this isn't homework, my semester is over.
this isn't homework, my semester is over.
math-chick_41 said:it expands to abab , but how come (ab)^-1 = (b^-1)(a^-1)?
A Non-Abelian Group is a mathematical structure that follows the group axioms, but does not satisfy the commutative property. This means that the order in which group elements are multiplied matters.
The equation (ab)^2 = (b^2)(a^2) is a special property of Non-Abelian Groups called the inverse square property. It states that the product of two group elements, when squared, is equal to the square of each element multiplied in reverse order.
The commutative property states that the order of multiplication does not matter, while the inverse square property of Non-Abelian Groups shows that the order of multiplication does matter.
No, not all Non-Abelian Groups have the inverse square property. This property is specific to certain types of Non-Abelian Groups, such as dihedral groups and quaternion groups.
Non-Abelian Groups with the inverse square property have applications in various fields such as physics, chemistry, and cryptography. They can be used to describe the symmetries of physical systems and molecules, and to create secure encryption algorithms.