Can Unique Subgroups Be Generalized as Normal Subgroups?

[brownnrl]

Our professor posed this question, and I'm having a very difficult time with it.

If you have a unique subgroup H of G such that H is of order 10 or 20, then H is a normal subgroup. How can you generalize this?

If you have the time to give some hints or suggestions, I'd appreciate it.

-Nelson

 

[brownnrl]

Quick note: By stating that theorem, he meant prove it for orders of 10 or 20, then generalize the findings.

 

[tacman]

It is not possible to generalize unique subgroups as normal subgroups. Normal subgroups are a specific type of subgroup that satisfy certain properties, such as being invariant under conjugation by elements of the larger group. Unique subgroups, on the other hand, are simply subgroups that have no other subgroups with the same order. These two concepts are not equivalent and cannot be generalized as one another.

Additionally, the statement given by the professor about unique subgroups of order 10 or 20 being normal is not necessarily true. While it may be the case for the specific example given, it is not a general rule. Normality is a property that must be proven for each subgroup individually, it cannot be assumed based on the subgroup's order alone.

In order to generalize unique subgroups as normal subgroups, you would need to prove that all unique subgroups in any group are normal. However, this is not true in general and therefore cannot be generalized. It is important to carefully consider the definitions and properties of both unique and normal subgroups in order to fully understand their differences and limitations.