Must a Non-Abelian Group of Order 10 Have an Element of Order 2?

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Homework Help Overview

The discussion revolves around the properties of non-abelian groups, specifically focusing on a group of order 10 and the existence of elements of certain orders, particularly order 2.

Discussion Character

  • Conceptual clarification, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the implications of Lagrange's theorem regarding the orders of elements in the group. There is a question about the possibility of all elements having order 5 and what that would mean for the group's structure.

Discussion Status

Some participants are seeking clarification on the application of Lagrange's theorem and its implications for the group's properties. There is a mix of attempts to understand the problem and questions about the initial steps needed to approach the proof.

Contextual Notes

There is mention of confusion regarding Lagrange's theorem and its application to the problem, indicating a potential gap in foundational understanding that may affect the discussion.

Bds_Css
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Homework Statement



Prove that a non-abelian group of order 10 must have an element of order 2.



What if the order of every element is 5?
Prove there are 5 elements of order 2.
 
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Where is your working? Have you attempted this?
 


honestly, I have no work because I don't know where to begin
 


Bds_Css said:
What if the order of every element is 5?
Prove there are 5 elements of order 2.

I don't know what you mean by this but for the first part use Lagrange's theorem deduce that there are 2 possible orders of elements. If you assume there is no element of order 2 prove that this means the group is abelian.
 


sorry,
it is all one problem


Prove that a nonabelian group of order 10 must have an element of order 2. What if the order of every element is 5? Prove there are 5 elements of order 2.


I am having trouble understanding Lag. THM.

Thanks again for your help
 


Lagranges theorem says that the order of the subgroup must divide the order of the group. The order of a cyclic group is prime. If you take any element of the group, you can make a cyclic subgroup generated by that element, so Lagrange says that the order of any element must divide the order of the group. The two possibilities you have for a non identity element are 2 and 5.
 

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