)=p but H is not a normal subgroup of G. So for some g in G Hg≠gH. I know I didn't do much, but is this the correct way to start? What to do now?A normal subgroup with prime index is a subgroup of a group that is closed under the group's operation and has a prime number as its index. This means that the number of cosets (distinct left or right cosets) of the subgroup in the group is equal to a prime number.
A normal subgroup with prime index is different from a regular subgroup because it has a prime number as its index, while a regular subgroup can have any positive integer as its index. Additionally, a normal subgroup with prime index is a special type of subgroup that has certain properties, such as being invariant under conjugation, which regular subgroups may not have.
The significance of having a prime index for a normal subgroup lies in its properties and its relationship with the group it is a part of. A normal subgroup with prime index is important in group theory and has applications in various areas of mathematics, including algebra and number theory.
To determine if a subgroup has a prime index, you can divide the order of the group by the order of the subgroup. If the result is a prime number, then the subgroup has a prime index. Additionally, you can also check if the subgroup meets the definition of a normal subgroup with prime index.
Some examples of groups with normal subgroups of prime index include cyclic groups, dihedral groups, and symmetric groups. These groups have subgroups with prime indices due to their specific structures and properties. Additionally, many other groups in abstract algebra and other areas of mathematics may also have normal subgroups with prime index.