Prove that p-group of order p^n is not simple

[vanckzhu]

Homework Statement

Show that, for p prime and n greater than or equal to 2, that every group of order Pp^n is not simple.

Homework Equations

N/A

The Attempt at a Solution

Hint given regarding the center. I know that the center is not trivial, but I guess I have to show that the center is not the entire group...so the center (which obviously is a subgroup of p^n) is a non-trivial normal subgroup?

Edit: LaTex doesn't work here, sad.

 

[Sethric]

Partially correct. Assuming the center is a non-trivial proper subgroup of your p-group, then it is, by definition, normal - meaning your p-group is not simple. And it is required to be non-trivial. However, your center need not be a proper subgroup. If your center is the entire p-group, implying the group is abelian, what is your next step? Cauchy's Theorem would be useful.*edit* LaTeX does work here, use [ t e x ] and [ / t e x ], without spaces. There are some packages not included, but the basics will work.