PsychonautQQ said:Homework Statement
If K is normal in G and |g| = n for some g in G, show that the order of Kg in G/K divides n.
Homework Equations
None
The Attempt at a Solution
Okay so I feel like I have a solution but I don't use all the information given so I'm trying to find holes in it...
g^n = 1
K = Kg^n = (Kg)^n. So it seems If we take Kg to the nth power we get it's identity. This means that the order of Kg must divide n.
What am I missing here?
A factor group, also known as a quotient group, is a mathematical group that is formed by taking a subgroup of a larger group and "factoring out" the elements in that subgroup to form a new group. The elements in the factor group are the cosets of the subgroup, and the operation in the factor group is defined by the operation of the larger group.
A subgroup is a subset of a group that also forms a group under the same operation. A factor group, on the other hand, is formed by taking a subgroup and "factoring out" its elements to create a new group. In other words, a subgroup is a smaller group within a larger group, while a factor group is a new group formed from the larger group and a subgroup.
Factor groups are important in modern algebra because they allow us to study the structure of a group by looking at its subgroups. By factoring out certain elements, we can simplify the group and gain a better understanding of its properties. Factor groups are also used in various applications, such as in cryptography and coding theory.
The order of a factor group is equal to the index of the subgroup being factored out. In other words, it is the number of distinct cosets in the factor group. This can be calculated by dividing the order of the larger group by the order of the subgroup.
Factor groups can be found in various areas of mathematics and science. For example, in chemistry, the symmetry of molecules can be represented by factor groups. In computer science, error-correcting codes can be represented by factor groups. In physics, the symmetry of physical systems can also be described using factor groups.