Does Every Group of Prime Power Order Have a Subgroup of Prime Order?

PhysicsUnderg
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Homework Statement


Let G be a group with pk elements, where p is a prime number and k is greater than or equal to 1. Prove that G has a subgroup of order p.

The Attempt at a Solution


I attempted to prove this by showing that the conditions for a set to be subgroup form a subgroup of order p. I have become lost, however, in my proof. Any direction would be helpful. :-)
 
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Hi PhysicsUnderg! :smile:

Take an element g of G. What is the order of g?
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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