What is the Proof for Finite Group with Elements of Order p?

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Homework Help Overview

The discussion revolves around a proof related to finite groups, specifically addressing the number of elements of a certain order, where the order is a prime number p. The original poster expresses confusion regarding the relationship between the number of elements of order p and the implications of Lagrange's Theorem.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of having an element of order p and question how many additional elements of order p must exist in the group. There are discussions about the nature of inverses and the structure of subgroups generated by elements of order p.

Discussion Status

Some participants have offered guidance on considering the subgroup generated by an element of order p and the distinct elements that arise from powers of that element. Others express ongoing confusion and plan to seek further clarification from their professor.

Contextual Notes

There is mention of a "special problem" assigned for the semester, indicating that participants may be under specific constraints or expectations regarding their approach to the problem.

MrsDebby
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Homework Statement


Suppose a finite group has exactly n elements of order p where p is prime. Prove that either n=0 or p divides n + 1.


Homework Equations



My professor says that this proof is similar to the proof of Lagrange's Theorem, in our Abstract Algebra book (Gallian).

The Attempt at a Solution



I am so lost with this question. It is a "special problem" we've been given to work on all semester. I have tried letting H represent a subgroup of the finite group, and using the elements of order p in the original group to form left cosets of H in the group. Not sure I really understand that. Not sure where that is getting me. I am not used to feeling so lost when tackling a problem.

Please...can anyone steer me in the right direction?

Thank you!
 
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Start dealing with this by assuming there is an element x of order p. How many more elements of order p does that force the group to have? Assuming that those are all of the elements of order p, then what's the relation between n and p. Now generalize.
 
thank you! I will work with that awhile...and let you know how it goes!
 
I am sorry, I am still stuck. If x has order p, then x inverse must also have order p, and must also be in the group. then that is 2 elements...unless x is it's own inverse. So I think I am totally missing something here? HELP!
Thank you...
 
MrsDebby said:
I am sorry, I am still stuck. If x has order p, then x inverse must also have order p, and must also be in the group. then that is 2 elements...unless x is it's own inverse. So I think I am totally missing something here? HELP!
Thank you...

x and x inverse aren't all. x generates a subgroup containing p elements. Try writing down a simple group of prime order like Z5. How many elements have order 5?
 
What is \left(x^q\right)^p for some integer q? Is there a positive integer r smaller than p for which \left(x^q\right)^r=e might be true for some positive integer q less than p? If x is of order p, how many distinct elements that are powers of x exist? How many of those are of order p? For how many elements y that are powers of x, is x also a power of y, and vice versa? Then, if another element z that is not a power of x is of order p, how many distinct elements that are powers of z exist, and how many of those are of order p? It is seen that every subgroup of order p has only the element e in common, and the element e has order 1. This gives a general formula for the possible numbers of elements with order p.
 
If x is of order p, there are p distinct elements of powers of x...right? <x> will have order p also. one of those will be e, so p-1 of them can have order p.
Z5 has 4 elements with order 5.

thinking...


sorry, I am usually not so dense..
 
Yes, now consider the case of multiple subgroups of order p, such as Z_5\times Z_5:

e A A A A
B C D E F
B E C F D
B D F C E
B F E D C
 
I'm sorry...I am still getting no where...

I do appreciate everyone's help, though. I will report if I make any progress. Going to talk to the professor tomorrow...this is so discouraging...
 

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