naoufelabs said:Please I have a problem with natural log for group set as follow:
a*b=eln(a)*ln(b)
1- Show that the group law * is associative and commutative
2- Show that the group law * accept an element e (Identity element)
Thank you !
naoufelabs said:I tested with Maple e^ln(a)*ln(b), then the result is a^ln(b).
so I tested with 2 different numbers, example:
3^ln(2) = 2^ln(3) ==> Gives the same result.
So I found that the e^ln(a)*ln(b) is commutative.
naoufelabs said:I tested with Maple e^ln(a)*ln(b), then the result is a^ln(b).
so I tested with 2 different numbers, example:
3^ln(2) = 2^ln(3) ==> Gives the same result.
So I found that the e^ln(a)*ln(b) is commutative.
A group in algebraic structure is a set of elements that follow a specific set of rules, known as the group axioms. These rules include closure, associativity, identity, and inverses. Groups are used to analyze and solve problems in abstract algebra and have many real-world applications.
The basic properties of a group include closure, associativity, identity, and inverses. Closure means that when two elements in the group are combined using the group operation, the result is also an element of the group. Associativity means that the order in which the group operation is performed does not affect the result. Identity means that there exists an element in the group that when combined with any other element using the group operation, gives back the original element. Inverses mean that for every element in the group, there exists another element that when combined using the group operation, gives the identity element.
A subgroup is a subset of a group that also follows the group axioms. In other words, a subgroup is a smaller group contained within a larger group. All subgroups must have the same group operation as the larger group, but they may have a different set of elements. The identity element of the subgroup must also be the same as the identity element of the larger group.
No, not all sets can be turned into groups. For a set to be a group, it must follow the group axioms. If a set does not follow these rules, it cannot be considered a group. Additionally, the group operation must be well-defined for all elements in the set. This means that combining two elements using the group operation must always give a unique result.
There are many examples of groups, including the integers under addition, the non-zero real numbers under multiplication, and the set of permutations of a finite set. Other examples include the symmetry group of a geometric shape, the set of matrices with non-zero determinant under multiplication, and the set of rotations of a cube. Groups can also be found in abstract algebraic structures such as rings, fields, and vector spaces.